Stochastic Maximum Principle for Partial Information Optimal Control Problem of Forward-Backward...
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Research Article Stochastic Maximum Principle for Partial Information Optimal Control Problem of Forward-Backward Systems Involving Classical and Impulse Controls Yan Wang, 1 Aimin Song, 1 and Enmin Feng 2 1 School of Science, Dalian Jiaotong University, Dalian 116028, China 2 School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China Correspondence should be addressed to Yan Wang; [email protected] Received 1 January 2014; Revised 28 March 2014; Accepted 29 March 2014; Published 15 April 2014 Academic Editor: Xiaojie Su Copyright © 2014 Yan Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the partial information classical and impulse controls problem of forward-backward systems driven by L´ evy processes, where the control variable consists of two components: the classical stochastic control and the impulse control; the information available to the controller is possibly less than the full information, that is, partial information. We derive a maximum principle to give the sufficient and necessary optimality conditions for the local critical points of the classical and impulse controls problem. As an application, we apply the maximum principle to a portfolio optimization problem with piecewise consumption processes and give its explicit solutions. 1. Introduction e classical and impulse controls problems have received considerable attention in recent years due to their wide applicability in different areas, such as optimal control of the exchange rate between different currencies (see, e.g., [1– 3]), optimal financing and dividend control problem of an insurance company facing fixed and proportional transaction costs (see, e.g., [4, 5]), stochastic differential game (see, e.g., [6]), and dynamic output feedback controller design problem (see, e.g., [7] and the references therein). In the existing literatures, the dynamic programming pri- nciple and the maximum principle are two main approaches in solving these problems. In dynamic programming principle, the classical and impulse controls can be solved by a verification theorem and the value function is a solution to some quasi-variational inequalities. However, the dynamic programming approach relies on the assumption that the controlled system is Marko- vian; see, for example, [8–10]. ere have been some pioneering works on deriving maximum principles for the classical and impulse controls problems. For example, Wu and Zhang [11] established max- imum principle for stochastic recursive optimal control problems involving impulse controls; Wu and Zhang [12] gave maximum principle for classical and impulse controls of forward-backward systems. In their control problems, the information available to the controller is full information. In many practical systems, the controller only gets partial information, instead of full information, such as delayed information (see, e.g., [13–16]). e partial information stochastic control problem is not of a Markovian type and hence cannot be solved by dynamic programming. As a result, maximum principles are established to solve the partial information stochastic control problem. ere is already a rich literature and versions of corresponding maximum prin- ciples for partial information control problems. For example, Baghery and Øksendal [17] derived the maximum principle for partial information stochastic control problem, where the stochastic system is described by stochastic differential equa- tions (SDE hereaſter). An and Øksendal [18] gave a maximum principle for the stochastic differential game under partial information. Øksendal and Sul` em [19] established maximum Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 452124, 8 pages http://dx.doi.org/10.1155/2014/452124
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Transcript of Stochastic Maximum Principle for Partial Information Optimal Control Problem of Forward-Backward...
Research ArticleStochastic Maximum Principle for Partial InformationOptimal Control Problem of Forward-Backward SystemsInvolving Classical and Impulse Controls
Yan Wang1 Aimin Song1 and Enmin Feng2
1 School of Science Dalian Jiaotong University Dalian 116028 China2 School of Mathematical Sciences Dalian University of Technology Dalian 116023 China
Correspondence should be addressed to Yan Wang wymath163com
Received 1 January 2014 Revised 28 March 2014 Accepted 29 March 2014 Published 15 April 2014
Academic Editor Xiaojie Su
Copyright copy 2014 Yan Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We study the partial information classical and impulse controls problem of forward-backward systems driven by Levy processeswhere the control variable consists of two components the classical stochastic control and the impulse control the informationavailable to the controller is possibly less than the full information that is partial information We derive a maximum principle togive the sufficient and necessary optimality conditions for the local critical points of the classical and impulse controls problem Asan application we apply the maximum principle to a portfolio optimization problem with piecewise consumption processes andgive its explicit solutions
1 Introduction
The classical and impulse controls problems have receivedconsiderable attention in recent years due to their wideapplicability in different areas such as optimal control ofthe exchange rate between different currencies (see eg [1ndash3]) optimal financing and dividend control problem of aninsurance company facing fixed and proportional transactioncosts (see eg [4 5]) stochastic differential game (see eg[6]) and dynamic output feedback controller design problem(see eg [7] and the references therein)
In the existing literatures the dynamic programming pri-nciple and the maximum principle are two main approachesin solving these problems
In dynamic programming principle the classical andimpulse controls can be solved by a verification theorem andthe value function is a solution to some quasi-variationalinequalities However the dynamic programming approachrelies on the assumption that the controlled system is Marko-vian see for example [8ndash10]
There have been some pioneering works on derivingmaximum principles for the classical and impulse controls
problems For example Wu and Zhang [11] established max-imum principle for stochastic recursive optimal controlproblems involving impulse controls Wu and Zhang [12]gave maximum principle for classical and impulse controlsof forward-backward systems In their control problems theinformation available to the controller is full information
In many practical systems the controller only gets partialinformation instead of full information such as delayedinformation (see eg [13ndash16]) The partial informationstochastic control problem is not of a Markovian type andhence cannot be solved by dynamic programming As aresultmaximumprinciples are established to solve the partialinformation stochastic control problem There is already arich literature and versions of correspondingmaximumprin-ciples for partial information control problems For exampleBaghery and Oslashksendal [17] derived the maximum principlefor partial information stochastic control problem where thestochastic system is described by stochastic differential equa-tions (SDE hereafter) An andOslashksendal [18] gave amaximumprinciple for the stochastic differential game under partialinformation Oslashksendal and Sulem [19] established maximum
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 452124 8 pageshttpdxdoiorg1011552014452124
2 Abstract and Applied Analysis
principles for stochastic control of forward-backward systemsdriven by Levy processes In their control problems thecontrol variable is just the classical stochastic control process119906(sdot) To the best of our knowledge there is no literatureon studying the maximum principle for partial informationclassical and impulse controls problems whichmotivates ourwork
In this paper we study classical and impulse controlsproblems of forward-backward systems where the stochasticsystems are represented by forward-backward SDEs drivenby Levy processes the control variable consists of two com-ponents the stochastic control 119906(sdot) and the impulse control120585(sdot) and the information available to the controller is possiblypartial information rather than full information Becauseof the non-Markovian nature of the partial information wecannot use dynamic programming principle to solve theproblems Instead we derive a maximum principle whichallows us to handle the partial information case
The similarmaximumprinciple is also studied byWu andZhang [11] in the complete information case and with theBrownian motion setting There are three main differencesbetween our paper and [11] Firstly we study the moregeneral cases the forward-backward system is driven by Levyprocesses and the information available to the controller ispartial information Secondly their proof differs from oursThey used convex perturbation technique to establish themaximum principle Thirdly they assumed the concavityconditions of Hamiltonian and utility functional to makethe necessary optimality conditions turn out to be sufficientHowever the concavity conditions may not hold in manyapplications Consequently in our maximum principle for-mulation we give the sufficient and necessary optimalityconditions for the local critical points instead of globaloptimums without the assumption of concavity condition
The paper is organized as follows in the next sectionwe formulate the partial information classical and impulsecontrols of the forward-backward system driven by Levyprocesses In Section 3 we derive the stochastic maximumprinciple for the considered classical and impulse controlsproblem In Section 4 we apply the general results obtainedin Section 3 to give the solutions of the example Finally weconclude this paper in Section 5
2 Problem Formulation
Let (ΩF F119905119905ge0 119875) be a filtered probability space and let
120578(sdot) be a Levy process defined on it Let 119861(119905) be an F119905-
Brownian motion and let (119889119905 119889119911) = 119873(119889119905 119889119911) minus ](119889119911)119889119905be compensated Poisson random measures independent of119861(119905) where ] is the Levy measure of Levy process 120578(119905) withjump measure119873 such that 119864[1205782
119894(119905)] lt infin for all 119905 119905 isin [0infin)
F119905119905ge0
is the filtration generated by 119861(119905) and (119889119905 119889119911) (asusual augmented with all the 119875-null sets) We refer to [8] formore information about Levy processes
Suppose that we are given a subfiltration G119905
sube F119905
representing the information available to the controller attime 119905 119905 isin [0 119879] It is remarked that the partial informationof classical and impulse controls is different from the classical
and impulse controls of delay systems where the statefunction is described by the solution of stochastic differentialdelay equation (see eg [20])
Let 120591119894 119894 ge 1 be a given sequence of increasing G
119905-sto-
pping times such that 120591119894uarr +infin At 120591
119894we are free to intervene
and give the system an impulse 120585119894isin R where 120585
119894is G120591119894-mea-
surable random variable We define impulse process 120585(119905) by
120585 (119905) = sum
119894ge1
1205851198941[120591119894 119879]
(119905) 119905 le 119879 (1)
It is worth noting that the assumption 120591119894uarr +infin implies that
at most finitely many impulses may occur on [0 119879]Now we consider the forward-backward systems involv-
ing classical and impulse controls Given 119886 isin R and 120583 isin R0
let 119887 [0 119879] times R times 119880 rarr R 120590 [0 119879] times R times 119880 rarr R120574 [0 119879] times R times 119880 times R
0rarr R 119892 [0 119879] times R times R times
R times 119880 rarr R 119862 [0 119879] rarr R and 119863 [0 119879] rarr R bemeasurable mappings119880 is a nonempty convex set ofRThenthe forward-backward systems are described by forward-backward SDEs in the unknown processes 119860(119905) 119883(119905) 119884(119905)and119870(119905) as follows
119889119860 (119905) = 119887 (119905 119860 (119905) 119906 (119905)) 119889119905 + 120590 (119905 119860 (119905) 119906 (119905)) 119889119861 (119905)
+ int
R0
120574 (119905 119860 (119905) 119906 (119905) 119911) (119889119905 119889119911) + 119862 (119905) 119889120585 (119905)
119889119883 (119905) = minus119892 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119906 (119905)) 119889119905
+ 119884 (119905) 119889119861 (119905) + int
R0
119870 (119905 119911) (119889119905 119889119911)
minus 119863 (119905) 119889120585 (119905)
119883 (119879) = 120583119860 (119879) 119860 (0) = 119886
(2)
The result of giving the impulse 120585119894is that the state jumps from
(119860(120591119894minus)119883(120591
119894minus)) to (119860(120591
119894) 119883(120591
119894)) = (119860(120591
119894minus)+119862(120591
119894)120585119894 119883(120591119894minus)minus
119863(120591119894)120585119894) We call (119906 120585) classical and impulse controls
There are two different jumps in the system (2) One jumpis the jump of (119860(120591) 119883(120591)) stemming from the randommea-sure119873 denoted by (Δ
119873119860(120591) Δ
119873119883(120591))The other jump is the
jump caused by the impulse 120585 given by (Δ120585119860(120591119894) Δ120585119883(120591119894)) =
(119862(120591119894)120585119894 119881(120591119894)120585119894) Let
M = (Δ119873119860 (120591) Δ
119873119883 (120591)) 0 le 120591 le 119879
N = (Δ120585119860 (120591119894) Δ120585119883(120591119894)) 0 le 120591
119894le 119879
(3)
Assumption 1 A jump (Δ119860(119905) Δ119883(119905)) at time 119905 0 le 119905 le 119879satisfies
(Δ119860 (119905) Δ119883 (119905)) isin M cupN
(Δ119860 (119905) Δ119883 (119905)) notin M capN
(4)
Let UG denote a given family of controls contained inthe set of G
119905-predictable controls 119906(sdot) such that the system
(2) has a unique strong solution We denote by I the class
Abstract and Applied Analysis 3
of processes 120585(sdot) = sum119894ge1
120585119894120594[120591119894 119879]
(sdot) such that each 120585119894is an R-
valuedG120591119894-measurable random variable LetKG be the class
of impulse process 120585 isin I such that119864(sum119894ge1
|120585119894|)2lt infinWe call
AG = UG timesKG the admissible control setSuppose we are given a performance functional of the
form
J (119906 120585) = 119864[int
119879
0
119891 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)) 119889119905
+ℎ1(119883 (0)) + ℎ
2(119860 (119879)) +sum
119894ge1
119897 (120591119894 120585119894)]
(5)
where 119864 denotes expectation with respect to 119875 and 119891 ℎ1 and
ℎ2are given functions such that
119864[int
119879
0
1003816100381610038161003816119891 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905))
1003816100381610038161003816119889119905
+1003816100381610038161003816ℎ1(119883 (0))
1003816100381610038161003816+1003816100381610038161003816ℎ2(119860 (119879))
1003816100381610038161003816+ sum
119894ge1
1003816100381610038161003816119897 (120591119894 120585119894)1003816100381610038161003816] lt infin
(6)
Then the classical and impulse controls problem is to find thevalue function ΦG(119886) isin R and optimal classical and impulsecontrols (119906lowast 120585lowast) isin AG such that
ΦG (119886) = sup(119906120585)isinAG
J (119906 120585) = J (119906lowast 120585lowast) (7)
3 Maximum Principle for Partial InformationClassical and Impulse Controls Problems
In this section we derive a maximum principle for theoptimal control problems (7) We will give the necessary andsufficient conditions for the local critical points (119906lowast 120585lowast)
Firstly we make the following assumptions
Assumption 2 (1) For all 119904 isin [0 119879) and bounded G119904-
measurable random variables 120579(120596) the control 120573119904defined by
120573119904(119905) = 120579 (120596) 120594
(119904119879] 119904 isin [0 119879] (8)
belongs toUG(2) For all (119906 120585) (120573 120589) isin AG where (120573 120589) is bounded
there exists 120575 gt 0 such that the control
(119906 (119905) + 119910120573 (119905) 120585 (119905) + 119910120589 (119905)) isin AG
forall119910 isin (minus120575 120575) 119905 isin [0 119879]
(9)
Next we give the definition of the Hamiltonian process
Definition 3 (see [19]) We define a Hamiltonian process
119867 [0 119879] timesR timesR timesR times 1198712(]) times 119880 timesR 997888rarr R (10)
as follows119867(119905 119886 119909 119910 119896 119906 120582)
= 119891 (119905 119886 119909 119910 119896 119906) + 120582 (119905) 119892 (119905 119886 119909 119910 119906)
+ 119887 (119905 119886 119906) 119901 (119905) + 120590 (119905 119886 119906) 119902 (119905)
+ int
R0
120574 (119905 119886 119906 119911) 119903 (119905 119911) ] (119889119911)
(11)
where 119867 is Frechet differentiable in the variables 119886 119909 119910 119896nabla119896119867 denotes the Frechet derivative in 119896 of 119867 the adjoint
processes 119901(119905) 119902(119905) 119903(119905 119911) and 120582(119905) are given by a pair offorward-backward SDEs as follows
(i) Forward system in the unknown process 120582(119905)
119889120582 (119905)
=
120597119867
120597119909
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot)
119906 (119905) 120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119905
+
120597119867
120597119910
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)
120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119861 (119905)
+ int
R0
nabla119896119867(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)
120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) (119889119905 119889119911)
120582 (0) = ℎ1015840
1(119883 (0))
(12)
(ii) Backward system in the unknown processes119901(119905) 119902(119905)and 119903(119905 sdot)
119889119901 (119905)
= minus
120597119867
120597119886
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905) 120582 (119905)
119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119905
+ 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = 120583120582 (119879) + ℎ1015840
2(119860 (119879))
(13)
For the sake of simplicity we use the short hand notationin the following
120597119887
120597119886
(119905 119860 (119905) 119906 (119905) 120596) =
120597119887
120597119886
(119905)
120597119887
120597119906
(119905 119860 (119905) 119906 (119905) 120596) =
120597119887
120597119906
(119905)
(14)
and similarly for (120597120590120597119886)(119905) (120597120590120597119906)(119905) (120597120574120597119886)(119905) (120597120574120597119906)(119905) (120597119891120597119886)(119905) (120597119891120597119909)(119905) (120597119891120597119910)(119905) (120597119891120597119906)(119905) nabla
119896119891(119905 119911)
(120597119892120597119886)(119905) (120597119892120597119909)(119905) (120597119892120597119910)(119905) and (120597119892120597119906)(119905)
4 Abstract and Applied Analysis
Theorem 4 (maximum principle) Let (119906 120585) isin AG withcorresponding solutions 119860(119905) 119883(119905) 119884(119905) 119870(119905 119911) and 120582(119905) of(2) (12) and (13) Assume that for all (119906 120585) isin AG the followinggrowth conditions hold
119864[int
119879
0
1198832(119905) ((
120597119867
120597119910
(119905))
2
+ int
R0
1003817100381710038171003817nabla119896119867(119905 119911)
1003817100381710038171003817
2] (119889119911)) 119889119905]lt infin
119864 [int
119879
0
1205822(119905) (119884
2(119905) + int
R0
1198702(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1198602(119905) (119902
2(119905) + int
R0
1199032(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1199012(119905) ((120590
2(119905))
2
+ int
R0
1205742(119905 119911) ] (119889119911)) 119889119905] lt infin
(15)
Then the following are equivalent(1) (119906 120585) is a critical point forJ(119906 120585) in the sense that
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
forall119887119900119906119899119889119890119889(120573 120589) isin AG
(16)
(2) Consider
119864[
120597
120597119906
119867(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 120582 (119905))119906=119906(119905)
|G119905] = 0
(17)
for aa (119905 120596) isin [0 119879] times Ω and
sum
120591119894le119879
119864[119901 (120591119894) 119862 (120591119894) +
120597119897
120597120585
(120591119894) minus 120582 (120591
119894)119863 (120591
119894) | G
120591119894] = 0
(18)
Proof Define
(119905 120573 120589) =
119889
119889119910
119860 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119883 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119884 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 119911 120573 120589) =
119889
119889119910
119870 (119905 119911 119906 + 119910120573 120585 + 119910120589) |119910=0
(19)
Then we have
(0 120573 120589) =
119889
119889119910
119860 (0 119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
(119879 120573 120589) =
119889
119889119910
119860 (119879 119906 + 119910120573 120585 + 119910120589) |119910=0
=
1
120583
(119879 120573 120589)
119889 (119905 120573 120589) = [
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)] 119889119905
+ int
119905
0
[
120597120590
120597119886
(119904) (119905 120573 120589) +
120597120590
120597119906
(119904) 120573 (119904)] 119889119861 (119904)
+ int
R0
[
120597120574
120597119886
(119905) (119905 120573 120589)+
120597120574
120597119906
(119905) 120573 (119905)](119889119905 119889119911)
+ 119862 (119905) 119889120589 (119905)
119889(119905 120573 120589) = minus [
120597119892
120597119886
(119905) (119905 120573 120589) +
120597119892
120597119909
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) (119905 120573 120589) +
120597119892
120597119906
(119905) 120573 (119905)] 119889119905
+ (119905 120573 120589) 119889119861 (119905)
+ int
R0
(119905 119911 120573 120589) (119889119905 119889119911) + 119863 (119905) 119889120589 (119905)
(20)
Firstly we prove (1) rArr (2) Assume that (1) holds Thenwe have
0 =
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 119864[
[
int
119879
0
120597119891
120597119886
(119905) (119905 120573 120589) +
120597119891
120597119909
(119905) (119905 120573 120589)
+
120597119891
120597119910
(119905) (119905 120573 120589)
+ int
R0
nabla119896119891 (119905 119911) (119905 119911 120573 120589) ] (119889119911)
+
120597119891
120597119906
(119905) 120573 (119905) 119889119905 + ℎ1015840
1(119883 (0)) (0 120573 120589)
+ ℎ1015840
2(119860 (119879)) (119879 120573 120589) + sum
120591119894le119879
120597119897
120597120578
(120591119894) 120589119894]
]
(21)
By Ito formula we get
119864 [ℎ1015840
1(119883 (0)) (0 120573 120589)]
= 119864 [120582 (0) (0 120573 120589)]
Abstract and Applied Analysis 5
= 119864[120582 (119879) (119879 120573 120589) minus int
119879
0
(119905 120573 120589)
120597119867
120597119909
(119905) 119889119905
minus int
119879
0
120597119867
120597119910
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
120582 (119905) (
120597119892
120597119886
(119905) (119905 120573 120589)
+
120597119892
120597119909
(119905) (119905 120573 120589) +
120597119892
120597119910
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) 120573 (119905)) 119889119905
minus int
119879
0
int
R0
nabla119896119867(119905 119911) (119905 119911 120573 120589) ] (119889119911) 119889119905
minussum
119894ge1
120582 (120591119894)119863 (120591
119894) 120589119894]
(22)
where 0 le 120591119894le 119879 Now we consider
119864 [ℎ1015840
2(119860 (119879)) (119879 120573 120589)]
= 119864 [(119901 (119879) minus 120583120582 (119879)) (119879 120573 120589)]
= 119864 [119901 (119879) (119879 120573 120589)] minus 119864 [120582 (119879) (119879 120573 120589)]
(23)
Applying Ito formula to 119864[119901(119879)(119879 120573 120589)] we get
119864 [119901 (119879) (119879 120573 120589)]
= 119864 [119901 (0) (0 120573 120589)
+ int
119879
0
119901 (119905) (
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)) 119889119905
+sum
119894ge1
119901 (120591119894) 119862 (120591119894) 120589119894minus int
119879
0
120597119867
120597119886
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
(
120597120590
120597119886
(119905) (119905 120573 120589) +
120597120590
120597119906
(119905) 120573 (119905)) 119902 (119905) 119889119905
+ int
119879
0
int
R0
119903 (119905 119911) (
120597120574
120597119886
(119905 119911) (119905 120573 120589)
+
120597120574
120597119906
(119905 119911) 120573 (119905)) ] (119889119911) 119889119905]
(24)
where 0 le 120591119894le 119879 By substituting (22) (23) and (24) into
(21) we obtain
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
(
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
minus
120597119867
120597119886
(119905)) (119905 120573 120589)
+ (
120597119891
120597119909
(119905) + 120582 (119905)
120597119892
120597119909
(119905) minus
120597119867
120597119909
(119905)) (119905 120573 120589)
+ (
120597119891
120597119910
(119905) + 120582 (119905)
120597119892
120597119910
(119905) minus
120597119867
120597119910
(119905)) (119905 120573 120589)
+ int
R0
(nabla119896119891 (119905 119911) minus nabla
119896119867(119905 119911)) (119905 119911 120573 120589) ] (119889119911)
+ (
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905)
+int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)) 120573 (119905) 119889119905]
(25)
Depending on the definition of Hamiltonian119867 we get
120597119867
120597119909
(119905) =
120597119891
120597119909
(119905) +
120597119892
120597119909
(119905) 120582 (119905)
120597119867
120597119910
(119905) =
120597119891
120597119910
(119905) +
120597119892
120597119910
(119905) 120582 (119905)
nabla119896119867(119905 119911) = nabla
119896119891 (119905 119911)
120597119867
120597119886
(119905) =
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
120597119867
120597119906
(119905) =
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905) + int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)
(26)
6 Abstract and Applied Analysis
Hence (25) simplifies to
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905]
(27)
for all bounded (120573 120589) isin AG It is obvious that 120573(119905) isindependent of 120589(119905) 0 le 119905 le 119879 So we obtain from (27) that
119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905] = 0 (28)
119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894] = 0
(29)
holds for all bounded 120573 isin UG and 120589 isin IGNow we prove that (17) holds for all 120573(119905) isin UG We know
that (28) holds for all bounded 120573 isin UG So (28) holds for allbounded 120573 isin UG of the form
120573 (119905) = 120573119904(119905 120596) = 120579 (120596) 120594
[119904119879](119905) 119905 isin [0 119879] (30)
for a fixed 119904 isin [0 119879) where 120579(120596) is a boundedG119904-measurable
random variable Then we have
119864[
120597119867
120597119906
(119904) 120579] = 0 (31)
which holds for all boundedG119904-measurable random variable
120579 As a result we conclude that
119864[
120597119867
120597119906
(119904)
10038161003816100381610038161003816100381610038161003816
G119904] = 0 (32)
Moreover since (29) holds for all bounded G120591119894-measurable
random variable 120589119894 we conclude that (18) holds Therefore
we conclude that (1) rArr (2)(2) rArr (1) Each bounded 120573 isin U
119866can be approximated by
linear combinations of controls 120573119904of the formThenwe prove
that (2) rArr (1) by reversing the above argument
Remark 5 Let 119909 rarr ℎ1(119909) 119886 rarr ℎ
2(119886) 120585 rarr 119897(119905 120585) and
(119905 119886 119909 119896 119906) rarr 119867(119905 119886 119909 119910 119896 119906 120582) be concave for all 119905 isin
[0 119879]Then the local critical point (119906 120585) which is obtained byTheorem 4 is also a global optimum for the control problem(7)
Remark 6 Let G119905= F119905 119870(119905 119911) = 0 and 120574(119905 119886 119906 119911) = 0
Then ourmaximumprinciple (Theorem 4) coincideswith themaximum principle (Theorem 31) in [11]
4 Application
Example 7 (portfolio optimization problem) In a financialmarket we are given a subfiltration
G119905sube F119905
forall119905 isin [0 119879] (33)
representing the information available to the trader at time 119905Let 120585(119905) = sum
119894ge11205851198941[120591119894 119879]
(119905) 119905 le 119879 be a piecewiseconsumption process (see eg [11]) where 120591
119894 is a fixed
sequence of increasing G119905-stopping times and each 120585
119894is an
G120591119894-measurable random variable Then the wealth process
119860(119905) = 119860119906120578(119905) corresponding to the portfolio 119906(119905) is given
by
119889119860 (119905) = 119906 (119905) [120577 (119905) 119889119905 + 120587 (119905) 119889119861 (119905)
+int
R0
984858 (119905 119911) (119889119905 119889119911)] minus 120603119889120585 (119905)
119860 (0) = 119886 gt 0
(34)
where 120603 ge 0 R0= R 0 and 120587(119905) and 984858(119905 119911) are F
119905-
predictable processes such that 984858(119905 119911) ge minus1+120598 for some 120598 gt 0
and
int
119879
0
1003816100381610038161003816120577 (119905)
1003816100381610038161003816+ 1205872(119905) + int
R0
9848582(119905 119911) ] (119889119911) 119889119905 lt infin as
(35)
Endowed with initial wealth 119886 gt 0 an investor wantsto find a portfolio strategy 119906(sdot) and a consumption strategy120585(sdot) minimizing an expected functional which composes ofthree parts the first part is the total utility of the consumptionminusint
119879
0(1199062(119905)2)119889119905 the second part represents the risk of the
terminal wealth 120588(119860(119879)) = 119883minus119860119906(119879)
119892(0) where 119883minus119860119906(119879)
119892(0)
is the value at 119905 = 0 of the solution 119883(119905) of the followingbackward stochastic differential equation ([19])
119889119883 (119905) = minus119892 (119905 119883 (119905)) 119889119905 + 119884 (119905) 119889119861 (119905)
+ int
R0
119870 (119905 119911) (119889119905 119889119911) minus 120599119889120585 (119905)
119883 (119879) = minus119860 (119879)
(36)
and the third part is the utility derived from the consump-tion process 120585(sdot) More precisely for any admissible control(119906(sdot) 120585(sdot)) the utility functional is defined by
119869 (119906 (sdot) 120585 (sdot)) = 119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
(37)
where119864 denotes the expectation with respect to the probabil-ity measure 119875 and 119878 gt 0 Therefore the control problem is tofind Φ(119886) and (119906lowast(sdot) 120585lowast(sdot)) such that
Φ (119886) = inf(119906120585)isinAG
119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
= 119864[
[
minusint
119879
0
119906lowast2(119905)
2
119889119905 + 119883minus119860119906lowast (119879)
119892(0) +
119878
2
sum
120591119894le119879
120585lowast2
119894]
]
(38)
Abstract and Applied Analysis 7
The control problem (38) is a classical and impulsecontrols problem of forward-backward systems driven byLevy processes under partial information G
119905 Next we solve
the control problem (38) byTheorem 4 With the notation ofthe previous section we see that in Example 7 we have
119891 (119905 119886 119909 119910 119896 119906 120596) = minus
1199062
2
ℎ1(119909) = 119909 ℎ
2(119904 120596) = 0
119887 (119905 119886 119906 120596) = 119906120577 (119905) 120590 (119905 119886 119906 120596) = 119906120587 (119905)
120574 (119905 119886 119906 119911 120596) = 119906984858 (119905 119911) 119897 (120591119894 120585119894) =
119878
2
1205852
119894
119862 (119905) = 120603 119863 (119905) = 120599 120583 = minus1
(39)
Then by (11) the Hamiltonian is
119867(119905 119886 119909 119910 119896 119906 120582 120596) = minus
1199062
2
+ 120582 (119905) 119892 (119905 119909)
+ 119906 (119905) 120577 (119905) 119901 (119905) + 119906 (119905) 120587 (119905) 119902 (119905)
+ int
R0
119906 (119905) 984858 (119905 119911) 119903 (119905 119911) ] (119889119911)
(40)
where
119889119901 (119905) = 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = minus 120582 (119879)
(41)
and 120582(119905) is given by (12) that is
119889120582 (119905) = 120582 (119905) 119892119909(119905 119883 (119905)) 119889119905
120582 (0) = 1
(42)
where 119892119909(119905 119909) = (120597120597119909)119892(119905 119909) We can easily obtain the
solution of (42) as follows
120582 (119905) = expint119905
0
119892119909(119904 119883 (119904)) 119889119904 0 le 119905 le 119879 (43)
If (119906lowast(119905) 120585lowast(119905)) is a local critical point with corresponding119883lowast(119905) = 119883
(119906lowast)(119905) then by the sufficient and necessary
optimality condition (17) in Theorem 4 we get
119864 [119906lowast(119905) | G
119905] = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(44)
Since 119906lowast(119905) isG119905-adapted we have
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(45)
where 119901(119905) 119902(119905) and 119903(119905 119911) are given by (41)
On the other hand by the sufficient and necessary opti-mality condition (17) in Theorem 4 we obtain
sum
120591119894lt119879
119864 [119878120585lowast
119894+ 120603119901 (120591
119894) minus 120599120582 (120591
119894) | G120591119894] = 0 (46)
That is for each 120591119894lt 119879 we have
119864 [120585lowast
119894| G120591119894] =
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (47)
Since 120585119894is anG
120591119894-measurable random variable we have
120585lowast
119894=
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (48)
where 120582(119905) is given by (43) and 119901(119905) is given by (41) Con-sequently we summarize the above results in the followingtheorem
Theorem 8 Let 119901(119905) 119902(119905) and 119903(119905 119911) be the solutions of (41)and let 120582(119905) be the solution of (43) Then the pair (119906lowast(119905) 120585lowast(119905))is given by
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
120585lowast(119905) = sum
119894ge1
120585lowast
1198941[120591119894 119879]
(119905) 119905 le 119879
(49)
where 120585lowast119894given by (48) is the local critical point of the classical
and impulse controls problem (38)
5 Conclusion
We consider the partial information classical and impulsecontrols problem of forward-backward systems driven byLevy processes The control variable consists of two com-ponents the classical stochastic control and the impulsecontrol Because of the non-Markovian nature of the par-tial information dynamic programming principle cannotbe used to solve partial information control problems Asa result we derive a maximum principle for this partialinformation problem Because the concavity conditions of theutility functions and the Hamiltonian process may not holdin many applications we give the sufficient and necessaryoptimality conditions for the local critical points of thecontrol problem To illustrate the theoretical results we usethe maximum principle to solve a portfolio optimizationproblem with piecewise consumption processes and give itsexplicit solutions
In this paper we assume that the two different jumps inour system do not occur at the same time (Assumption 1)This assumption makes the problem easier to analyze How-ever it may fail in many applications Without this assump-tion it requiresmore attention to distinguish between the twodifferent jumpsThiswill be explored in our subsequentwork
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
principles for stochastic control of forward-backward systemsdriven by Levy processes In their control problems thecontrol variable is just the classical stochastic control process119906(sdot) To the best of our knowledge there is no literatureon studying the maximum principle for partial informationclassical and impulse controls problems whichmotivates ourwork
In this paper we study classical and impulse controlsproblems of forward-backward systems where the stochasticsystems are represented by forward-backward SDEs drivenby Levy processes the control variable consists of two com-ponents the stochastic control 119906(sdot) and the impulse control120585(sdot) and the information available to the controller is possiblypartial information rather than full information Becauseof the non-Markovian nature of the partial information wecannot use dynamic programming principle to solve theproblems Instead we derive a maximum principle whichallows us to handle the partial information case
The similarmaximumprinciple is also studied byWu andZhang [11] in the complete information case and with theBrownian motion setting There are three main differencesbetween our paper and [11] Firstly we study the moregeneral cases the forward-backward system is driven by Levyprocesses and the information available to the controller ispartial information Secondly their proof differs from oursThey used convex perturbation technique to establish themaximum principle Thirdly they assumed the concavityconditions of Hamiltonian and utility functional to makethe necessary optimality conditions turn out to be sufficientHowever the concavity conditions may not hold in manyapplications Consequently in our maximum principle for-mulation we give the sufficient and necessary optimalityconditions for the local critical points instead of globaloptimums without the assumption of concavity condition
The paper is organized as follows in the next sectionwe formulate the partial information classical and impulsecontrols of the forward-backward system driven by Levyprocesses In Section 3 we derive the stochastic maximumprinciple for the considered classical and impulse controlsproblem In Section 4 we apply the general results obtainedin Section 3 to give the solutions of the example Finally weconclude this paper in Section 5
2 Problem Formulation
Let (ΩF F119905119905ge0 119875) be a filtered probability space and let
120578(sdot) be a Levy process defined on it Let 119861(119905) be an F119905-
Brownian motion and let (119889119905 119889119911) = 119873(119889119905 119889119911) minus ](119889119911)119889119905be compensated Poisson random measures independent of119861(119905) where ] is the Levy measure of Levy process 120578(119905) withjump measure119873 such that 119864[1205782
119894(119905)] lt infin for all 119905 119905 isin [0infin)
F119905119905ge0
is the filtration generated by 119861(119905) and (119889119905 119889119911) (asusual augmented with all the 119875-null sets) We refer to [8] formore information about Levy processes
Suppose that we are given a subfiltration G119905
sube F119905
representing the information available to the controller attime 119905 119905 isin [0 119879] It is remarked that the partial informationof classical and impulse controls is different from the classical
and impulse controls of delay systems where the statefunction is described by the solution of stochastic differentialdelay equation (see eg [20])
Let 120591119894 119894 ge 1 be a given sequence of increasing G
119905-sto-
pping times such that 120591119894uarr +infin At 120591
119894we are free to intervene
and give the system an impulse 120585119894isin R where 120585
119894is G120591119894-mea-
surable random variable We define impulse process 120585(119905) by
120585 (119905) = sum
119894ge1
1205851198941[120591119894 119879]
(119905) 119905 le 119879 (1)
It is worth noting that the assumption 120591119894uarr +infin implies that
at most finitely many impulses may occur on [0 119879]Now we consider the forward-backward systems involv-
ing classical and impulse controls Given 119886 isin R and 120583 isin R0
let 119887 [0 119879] times R times 119880 rarr R 120590 [0 119879] times R times 119880 rarr R120574 [0 119879] times R times 119880 times R
0rarr R 119892 [0 119879] times R times R times
R times 119880 rarr R 119862 [0 119879] rarr R and 119863 [0 119879] rarr R bemeasurable mappings119880 is a nonempty convex set ofRThenthe forward-backward systems are described by forward-backward SDEs in the unknown processes 119860(119905) 119883(119905) 119884(119905)and119870(119905) as follows
119889119860 (119905) = 119887 (119905 119860 (119905) 119906 (119905)) 119889119905 + 120590 (119905 119860 (119905) 119906 (119905)) 119889119861 (119905)
+ int
R0
120574 (119905 119860 (119905) 119906 (119905) 119911) (119889119905 119889119911) + 119862 (119905) 119889120585 (119905)
119889119883 (119905) = minus119892 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119906 (119905)) 119889119905
+ 119884 (119905) 119889119861 (119905) + int
R0
119870 (119905 119911) (119889119905 119889119911)
minus 119863 (119905) 119889120585 (119905)
119883 (119879) = 120583119860 (119879) 119860 (0) = 119886
(2)
The result of giving the impulse 120585119894is that the state jumps from
(119860(120591119894minus)119883(120591
119894minus)) to (119860(120591
119894) 119883(120591
119894)) = (119860(120591
119894minus)+119862(120591
119894)120585119894 119883(120591119894minus)minus
119863(120591119894)120585119894) We call (119906 120585) classical and impulse controls
There are two different jumps in the system (2) One jumpis the jump of (119860(120591) 119883(120591)) stemming from the randommea-sure119873 denoted by (Δ
119873119860(120591) Δ
119873119883(120591))The other jump is the
jump caused by the impulse 120585 given by (Δ120585119860(120591119894) Δ120585119883(120591119894)) =
(119862(120591119894)120585119894 119881(120591119894)120585119894) Let
M = (Δ119873119860 (120591) Δ
119873119883 (120591)) 0 le 120591 le 119879
N = (Δ120585119860 (120591119894) Δ120585119883(120591119894)) 0 le 120591
119894le 119879
(3)
Assumption 1 A jump (Δ119860(119905) Δ119883(119905)) at time 119905 0 le 119905 le 119879satisfies
(Δ119860 (119905) Δ119883 (119905)) isin M cupN
(Δ119860 (119905) Δ119883 (119905)) notin M capN
(4)
Let UG denote a given family of controls contained inthe set of G
119905-predictable controls 119906(sdot) such that the system
(2) has a unique strong solution We denote by I the class
Abstract and Applied Analysis 3
of processes 120585(sdot) = sum119894ge1
120585119894120594[120591119894 119879]
(sdot) such that each 120585119894is an R-
valuedG120591119894-measurable random variable LetKG be the class
of impulse process 120585 isin I such that119864(sum119894ge1
|120585119894|)2lt infinWe call
AG = UG timesKG the admissible control setSuppose we are given a performance functional of the
form
J (119906 120585) = 119864[int
119879
0
119891 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)) 119889119905
+ℎ1(119883 (0)) + ℎ
2(119860 (119879)) +sum
119894ge1
119897 (120591119894 120585119894)]
(5)
where 119864 denotes expectation with respect to 119875 and 119891 ℎ1 and
ℎ2are given functions such that
119864[int
119879
0
1003816100381610038161003816119891 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905))
1003816100381610038161003816119889119905
+1003816100381610038161003816ℎ1(119883 (0))
1003816100381610038161003816+1003816100381610038161003816ℎ2(119860 (119879))
1003816100381610038161003816+ sum
119894ge1
1003816100381610038161003816119897 (120591119894 120585119894)1003816100381610038161003816] lt infin
(6)
Then the classical and impulse controls problem is to find thevalue function ΦG(119886) isin R and optimal classical and impulsecontrols (119906lowast 120585lowast) isin AG such that
ΦG (119886) = sup(119906120585)isinAG
J (119906 120585) = J (119906lowast 120585lowast) (7)
3 Maximum Principle for Partial InformationClassical and Impulse Controls Problems
In this section we derive a maximum principle for theoptimal control problems (7) We will give the necessary andsufficient conditions for the local critical points (119906lowast 120585lowast)
Firstly we make the following assumptions
Assumption 2 (1) For all 119904 isin [0 119879) and bounded G119904-
measurable random variables 120579(120596) the control 120573119904defined by
120573119904(119905) = 120579 (120596) 120594
(119904119879] 119904 isin [0 119879] (8)
belongs toUG(2) For all (119906 120585) (120573 120589) isin AG where (120573 120589) is bounded
there exists 120575 gt 0 such that the control
(119906 (119905) + 119910120573 (119905) 120585 (119905) + 119910120589 (119905)) isin AG
forall119910 isin (minus120575 120575) 119905 isin [0 119879]
(9)
Next we give the definition of the Hamiltonian process
Definition 3 (see [19]) We define a Hamiltonian process
119867 [0 119879] timesR timesR timesR times 1198712(]) times 119880 timesR 997888rarr R (10)
as follows119867(119905 119886 119909 119910 119896 119906 120582)
= 119891 (119905 119886 119909 119910 119896 119906) + 120582 (119905) 119892 (119905 119886 119909 119910 119906)
+ 119887 (119905 119886 119906) 119901 (119905) + 120590 (119905 119886 119906) 119902 (119905)
+ int
R0
120574 (119905 119886 119906 119911) 119903 (119905 119911) ] (119889119911)
(11)
where 119867 is Frechet differentiable in the variables 119886 119909 119910 119896nabla119896119867 denotes the Frechet derivative in 119896 of 119867 the adjoint
processes 119901(119905) 119902(119905) 119903(119905 119911) and 120582(119905) are given by a pair offorward-backward SDEs as follows
(i) Forward system in the unknown process 120582(119905)
119889120582 (119905)
=
120597119867
120597119909
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot)
119906 (119905) 120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119905
+
120597119867
120597119910
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)
120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119861 (119905)
+ int
R0
nabla119896119867(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)
120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) (119889119905 119889119911)
120582 (0) = ℎ1015840
1(119883 (0))
(12)
(ii) Backward system in the unknown processes119901(119905) 119902(119905)and 119903(119905 sdot)
119889119901 (119905)
= minus
120597119867
120597119886
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905) 120582 (119905)
119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119905
+ 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = 120583120582 (119879) + ℎ1015840
2(119860 (119879))
(13)
For the sake of simplicity we use the short hand notationin the following
120597119887
120597119886
(119905 119860 (119905) 119906 (119905) 120596) =
120597119887
120597119886
(119905)
120597119887
120597119906
(119905 119860 (119905) 119906 (119905) 120596) =
120597119887
120597119906
(119905)
(14)
and similarly for (120597120590120597119886)(119905) (120597120590120597119906)(119905) (120597120574120597119886)(119905) (120597120574120597119906)(119905) (120597119891120597119886)(119905) (120597119891120597119909)(119905) (120597119891120597119910)(119905) (120597119891120597119906)(119905) nabla
119896119891(119905 119911)
(120597119892120597119886)(119905) (120597119892120597119909)(119905) (120597119892120597119910)(119905) and (120597119892120597119906)(119905)
4 Abstract and Applied Analysis
Theorem 4 (maximum principle) Let (119906 120585) isin AG withcorresponding solutions 119860(119905) 119883(119905) 119884(119905) 119870(119905 119911) and 120582(119905) of(2) (12) and (13) Assume that for all (119906 120585) isin AG the followinggrowth conditions hold
119864[int
119879
0
1198832(119905) ((
120597119867
120597119910
(119905))
2
+ int
R0
1003817100381710038171003817nabla119896119867(119905 119911)
1003817100381710038171003817
2] (119889119911)) 119889119905]lt infin
119864 [int
119879
0
1205822(119905) (119884
2(119905) + int
R0
1198702(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1198602(119905) (119902
2(119905) + int
R0
1199032(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1199012(119905) ((120590
2(119905))
2
+ int
R0
1205742(119905 119911) ] (119889119911)) 119889119905] lt infin
(15)
Then the following are equivalent(1) (119906 120585) is a critical point forJ(119906 120585) in the sense that
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
forall119887119900119906119899119889119890119889(120573 120589) isin AG
(16)
(2) Consider
119864[
120597
120597119906
119867(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 120582 (119905))119906=119906(119905)
|G119905] = 0
(17)
for aa (119905 120596) isin [0 119879] times Ω and
sum
120591119894le119879
119864[119901 (120591119894) 119862 (120591119894) +
120597119897
120597120585
(120591119894) minus 120582 (120591
119894)119863 (120591
119894) | G
120591119894] = 0
(18)
Proof Define
(119905 120573 120589) =
119889
119889119910
119860 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119883 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119884 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 119911 120573 120589) =
119889
119889119910
119870 (119905 119911 119906 + 119910120573 120585 + 119910120589) |119910=0
(19)
Then we have
(0 120573 120589) =
119889
119889119910
119860 (0 119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
(119879 120573 120589) =
119889
119889119910
119860 (119879 119906 + 119910120573 120585 + 119910120589) |119910=0
=
1
120583
(119879 120573 120589)
119889 (119905 120573 120589) = [
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)] 119889119905
+ int
119905
0
[
120597120590
120597119886
(119904) (119905 120573 120589) +
120597120590
120597119906
(119904) 120573 (119904)] 119889119861 (119904)
+ int
R0
[
120597120574
120597119886
(119905) (119905 120573 120589)+
120597120574
120597119906
(119905) 120573 (119905)](119889119905 119889119911)
+ 119862 (119905) 119889120589 (119905)
119889(119905 120573 120589) = minus [
120597119892
120597119886
(119905) (119905 120573 120589) +
120597119892
120597119909
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) (119905 120573 120589) +
120597119892
120597119906
(119905) 120573 (119905)] 119889119905
+ (119905 120573 120589) 119889119861 (119905)
+ int
R0
(119905 119911 120573 120589) (119889119905 119889119911) + 119863 (119905) 119889120589 (119905)
(20)
Firstly we prove (1) rArr (2) Assume that (1) holds Thenwe have
0 =
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 119864[
[
int
119879
0
120597119891
120597119886
(119905) (119905 120573 120589) +
120597119891
120597119909
(119905) (119905 120573 120589)
+
120597119891
120597119910
(119905) (119905 120573 120589)
+ int
R0
nabla119896119891 (119905 119911) (119905 119911 120573 120589) ] (119889119911)
+
120597119891
120597119906
(119905) 120573 (119905) 119889119905 + ℎ1015840
1(119883 (0)) (0 120573 120589)
+ ℎ1015840
2(119860 (119879)) (119879 120573 120589) + sum
120591119894le119879
120597119897
120597120578
(120591119894) 120589119894]
]
(21)
By Ito formula we get
119864 [ℎ1015840
1(119883 (0)) (0 120573 120589)]
= 119864 [120582 (0) (0 120573 120589)]
Abstract and Applied Analysis 5
= 119864[120582 (119879) (119879 120573 120589) minus int
119879
0
(119905 120573 120589)
120597119867
120597119909
(119905) 119889119905
minus int
119879
0
120597119867
120597119910
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
120582 (119905) (
120597119892
120597119886
(119905) (119905 120573 120589)
+
120597119892
120597119909
(119905) (119905 120573 120589) +
120597119892
120597119910
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) 120573 (119905)) 119889119905
minus int
119879
0
int
R0
nabla119896119867(119905 119911) (119905 119911 120573 120589) ] (119889119911) 119889119905
minussum
119894ge1
120582 (120591119894)119863 (120591
119894) 120589119894]
(22)
where 0 le 120591119894le 119879 Now we consider
119864 [ℎ1015840
2(119860 (119879)) (119879 120573 120589)]
= 119864 [(119901 (119879) minus 120583120582 (119879)) (119879 120573 120589)]
= 119864 [119901 (119879) (119879 120573 120589)] minus 119864 [120582 (119879) (119879 120573 120589)]
(23)
Applying Ito formula to 119864[119901(119879)(119879 120573 120589)] we get
119864 [119901 (119879) (119879 120573 120589)]
= 119864 [119901 (0) (0 120573 120589)
+ int
119879
0
119901 (119905) (
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)) 119889119905
+sum
119894ge1
119901 (120591119894) 119862 (120591119894) 120589119894minus int
119879
0
120597119867
120597119886
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
(
120597120590
120597119886
(119905) (119905 120573 120589) +
120597120590
120597119906
(119905) 120573 (119905)) 119902 (119905) 119889119905
+ int
119879
0
int
R0
119903 (119905 119911) (
120597120574
120597119886
(119905 119911) (119905 120573 120589)
+
120597120574
120597119906
(119905 119911) 120573 (119905)) ] (119889119911) 119889119905]
(24)
where 0 le 120591119894le 119879 By substituting (22) (23) and (24) into
(21) we obtain
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
(
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
minus
120597119867
120597119886
(119905)) (119905 120573 120589)
+ (
120597119891
120597119909
(119905) + 120582 (119905)
120597119892
120597119909
(119905) minus
120597119867
120597119909
(119905)) (119905 120573 120589)
+ (
120597119891
120597119910
(119905) + 120582 (119905)
120597119892
120597119910
(119905) minus
120597119867
120597119910
(119905)) (119905 120573 120589)
+ int
R0
(nabla119896119891 (119905 119911) minus nabla
119896119867(119905 119911)) (119905 119911 120573 120589) ] (119889119911)
+ (
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905)
+int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)) 120573 (119905) 119889119905]
(25)
Depending on the definition of Hamiltonian119867 we get
120597119867
120597119909
(119905) =
120597119891
120597119909
(119905) +
120597119892
120597119909
(119905) 120582 (119905)
120597119867
120597119910
(119905) =
120597119891
120597119910
(119905) +
120597119892
120597119910
(119905) 120582 (119905)
nabla119896119867(119905 119911) = nabla
119896119891 (119905 119911)
120597119867
120597119886
(119905) =
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
120597119867
120597119906
(119905) =
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905) + int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)
(26)
6 Abstract and Applied Analysis
Hence (25) simplifies to
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905]
(27)
for all bounded (120573 120589) isin AG It is obvious that 120573(119905) isindependent of 120589(119905) 0 le 119905 le 119879 So we obtain from (27) that
119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905] = 0 (28)
119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894] = 0
(29)
holds for all bounded 120573 isin UG and 120589 isin IGNow we prove that (17) holds for all 120573(119905) isin UG We know
that (28) holds for all bounded 120573 isin UG So (28) holds for allbounded 120573 isin UG of the form
120573 (119905) = 120573119904(119905 120596) = 120579 (120596) 120594
[119904119879](119905) 119905 isin [0 119879] (30)
for a fixed 119904 isin [0 119879) where 120579(120596) is a boundedG119904-measurable
random variable Then we have
119864[
120597119867
120597119906
(119904) 120579] = 0 (31)
which holds for all boundedG119904-measurable random variable
120579 As a result we conclude that
119864[
120597119867
120597119906
(119904)
10038161003816100381610038161003816100381610038161003816
G119904] = 0 (32)
Moreover since (29) holds for all bounded G120591119894-measurable
random variable 120589119894 we conclude that (18) holds Therefore
we conclude that (1) rArr (2)(2) rArr (1) Each bounded 120573 isin U
119866can be approximated by
linear combinations of controls 120573119904of the formThenwe prove
that (2) rArr (1) by reversing the above argument
Remark 5 Let 119909 rarr ℎ1(119909) 119886 rarr ℎ
2(119886) 120585 rarr 119897(119905 120585) and
(119905 119886 119909 119896 119906) rarr 119867(119905 119886 119909 119910 119896 119906 120582) be concave for all 119905 isin
[0 119879]Then the local critical point (119906 120585) which is obtained byTheorem 4 is also a global optimum for the control problem(7)
Remark 6 Let G119905= F119905 119870(119905 119911) = 0 and 120574(119905 119886 119906 119911) = 0
Then ourmaximumprinciple (Theorem 4) coincideswith themaximum principle (Theorem 31) in [11]
4 Application
Example 7 (portfolio optimization problem) In a financialmarket we are given a subfiltration
G119905sube F119905
forall119905 isin [0 119879] (33)
representing the information available to the trader at time 119905Let 120585(119905) = sum
119894ge11205851198941[120591119894 119879]
(119905) 119905 le 119879 be a piecewiseconsumption process (see eg [11]) where 120591
119894 is a fixed
sequence of increasing G119905-stopping times and each 120585
119894is an
G120591119894-measurable random variable Then the wealth process
119860(119905) = 119860119906120578(119905) corresponding to the portfolio 119906(119905) is given
by
119889119860 (119905) = 119906 (119905) [120577 (119905) 119889119905 + 120587 (119905) 119889119861 (119905)
+int
R0
984858 (119905 119911) (119889119905 119889119911)] minus 120603119889120585 (119905)
119860 (0) = 119886 gt 0
(34)
where 120603 ge 0 R0= R 0 and 120587(119905) and 984858(119905 119911) are F
119905-
predictable processes such that 984858(119905 119911) ge minus1+120598 for some 120598 gt 0
and
int
119879
0
1003816100381610038161003816120577 (119905)
1003816100381610038161003816+ 1205872(119905) + int
R0
9848582(119905 119911) ] (119889119911) 119889119905 lt infin as
(35)
Endowed with initial wealth 119886 gt 0 an investor wantsto find a portfolio strategy 119906(sdot) and a consumption strategy120585(sdot) minimizing an expected functional which composes ofthree parts the first part is the total utility of the consumptionminusint
119879
0(1199062(119905)2)119889119905 the second part represents the risk of the
terminal wealth 120588(119860(119879)) = 119883minus119860119906(119879)
119892(0) where 119883minus119860119906(119879)
119892(0)
is the value at 119905 = 0 of the solution 119883(119905) of the followingbackward stochastic differential equation ([19])
119889119883 (119905) = minus119892 (119905 119883 (119905)) 119889119905 + 119884 (119905) 119889119861 (119905)
+ int
R0
119870 (119905 119911) (119889119905 119889119911) minus 120599119889120585 (119905)
119883 (119879) = minus119860 (119879)
(36)
and the third part is the utility derived from the consump-tion process 120585(sdot) More precisely for any admissible control(119906(sdot) 120585(sdot)) the utility functional is defined by
119869 (119906 (sdot) 120585 (sdot)) = 119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
(37)
where119864 denotes the expectation with respect to the probabil-ity measure 119875 and 119878 gt 0 Therefore the control problem is tofind Φ(119886) and (119906lowast(sdot) 120585lowast(sdot)) such that
Φ (119886) = inf(119906120585)isinAG
119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
= 119864[
[
minusint
119879
0
119906lowast2(119905)
2
119889119905 + 119883minus119860119906lowast (119879)
119892(0) +
119878
2
sum
120591119894le119879
120585lowast2
119894]
]
(38)
Abstract and Applied Analysis 7
The control problem (38) is a classical and impulsecontrols problem of forward-backward systems driven byLevy processes under partial information G
119905 Next we solve
the control problem (38) byTheorem 4 With the notation ofthe previous section we see that in Example 7 we have
119891 (119905 119886 119909 119910 119896 119906 120596) = minus
1199062
2
ℎ1(119909) = 119909 ℎ
2(119904 120596) = 0
119887 (119905 119886 119906 120596) = 119906120577 (119905) 120590 (119905 119886 119906 120596) = 119906120587 (119905)
120574 (119905 119886 119906 119911 120596) = 119906984858 (119905 119911) 119897 (120591119894 120585119894) =
119878
2
1205852
119894
119862 (119905) = 120603 119863 (119905) = 120599 120583 = minus1
(39)
Then by (11) the Hamiltonian is
119867(119905 119886 119909 119910 119896 119906 120582 120596) = minus
1199062
2
+ 120582 (119905) 119892 (119905 119909)
+ 119906 (119905) 120577 (119905) 119901 (119905) + 119906 (119905) 120587 (119905) 119902 (119905)
+ int
R0
119906 (119905) 984858 (119905 119911) 119903 (119905 119911) ] (119889119911)
(40)
where
119889119901 (119905) = 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = minus 120582 (119879)
(41)
and 120582(119905) is given by (12) that is
119889120582 (119905) = 120582 (119905) 119892119909(119905 119883 (119905)) 119889119905
120582 (0) = 1
(42)
where 119892119909(119905 119909) = (120597120597119909)119892(119905 119909) We can easily obtain the
solution of (42) as follows
120582 (119905) = expint119905
0
119892119909(119904 119883 (119904)) 119889119904 0 le 119905 le 119879 (43)
If (119906lowast(119905) 120585lowast(119905)) is a local critical point with corresponding119883lowast(119905) = 119883
(119906lowast)(119905) then by the sufficient and necessary
optimality condition (17) in Theorem 4 we get
119864 [119906lowast(119905) | G
119905] = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(44)
Since 119906lowast(119905) isG119905-adapted we have
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(45)
where 119901(119905) 119902(119905) and 119903(119905 119911) are given by (41)
On the other hand by the sufficient and necessary opti-mality condition (17) in Theorem 4 we obtain
sum
120591119894lt119879
119864 [119878120585lowast
119894+ 120603119901 (120591
119894) minus 120599120582 (120591
119894) | G120591119894] = 0 (46)
That is for each 120591119894lt 119879 we have
119864 [120585lowast
119894| G120591119894] =
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (47)
Since 120585119894is anG
120591119894-measurable random variable we have
120585lowast
119894=
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (48)
where 120582(119905) is given by (43) and 119901(119905) is given by (41) Con-sequently we summarize the above results in the followingtheorem
Theorem 8 Let 119901(119905) 119902(119905) and 119903(119905 119911) be the solutions of (41)and let 120582(119905) be the solution of (43) Then the pair (119906lowast(119905) 120585lowast(119905))is given by
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
120585lowast(119905) = sum
119894ge1
120585lowast
1198941[120591119894 119879]
(119905) 119905 le 119879
(49)
where 120585lowast119894given by (48) is the local critical point of the classical
and impulse controls problem (38)
5 Conclusion
We consider the partial information classical and impulsecontrols problem of forward-backward systems driven byLevy processes The control variable consists of two com-ponents the classical stochastic control and the impulsecontrol Because of the non-Markovian nature of the par-tial information dynamic programming principle cannotbe used to solve partial information control problems Asa result we derive a maximum principle for this partialinformation problem Because the concavity conditions of theutility functions and the Hamiltonian process may not holdin many applications we give the sufficient and necessaryoptimality conditions for the local critical points of thecontrol problem To illustrate the theoretical results we usethe maximum principle to solve a portfolio optimizationproblem with piecewise consumption processes and give itsexplicit solutions
In this paper we assume that the two different jumps inour system do not occur at the same time (Assumption 1)This assumption makes the problem easier to analyze How-ever it may fail in many applications Without this assump-tion it requiresmore attention to distinguish between the twodifferent jumpsThiswill be explored in our subsequentwork
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
of processes 120585(sdot) = sum119894ge1
120585119894120594[120591119894 119879]
(sdot) such that each 120585119894is an R-
valuedG120591119894-measurable random variable LetKG be the class
of impulse process 120585 isin I such that119864(sum119894ge1
|120585119894|)2lt infinWe call
AG = UG timesKG the admissible control setSuppose we are given a performance functional of the
form
J (119906 120585) = 119864[int
119879
0
119891 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)) 119889119905
+ℎ1(119883 (0)) + ℎ
2(119860 (119879)) +sum
119894ge1
119897 (120591119894 120585119894)]
(5)
where 119864 denotes expectation with respect to 119875 and 119891 ℎ1 and
ℎ2are given functions such that
119864[int
119879
0
1003816100381610038161003816119891 (119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905))
1003816100381610038161003816119889119905
+1003816100381610038161003816ℎ1(119883 (0))
1003816100381610038161003816+1003816100381610038161003816ℎ2(119860 (119879))
1003816100381610038161003816+ sum
119894ge1
1003816100381610038161003816119897 (120591119894 120585119894)1003816100381610038161003816] lt infin
(6)
Then the classical and impulse controls problem is to find thevalue function ΦG(119886) isin R and optimal classical and impulsecontrols (119906lowast 120585lowast) isin AG such that
ΦG (119886) = sup(119906120585)isinAG
J (119906 120585) = J (119906lowast 120585lowast) (7)
3 Maximum Principle for Partial InformationClassical and Impulse Controls Problems
In this section we derive a maximum principle for theoptimal control problems (7) We will give the necessary andsufficient conditions for the local critical points (119906lowast 120585lowast)
Firstly we make the following assumptions
Assumption 2 (1) For all 119904 isin [0 119879) and bounded G119904-
measurable random variables 120579(120596) the control 120573119904defined by
120573119904(119905) = 120579 (120596) 120594
(119904119879] 119904 isin [0 119879] (8)
belongs toUG(2) For all (119906 120585) (120573 120589) isin AG where (120573 120589) is bounded
there exists 120575 gt 0 such that the control
(119906 (119905) + 119910120573 (119905) 120585 (119905) + 119910120589 (119905)) isin AG
forall119910 isin (minus120575 120575) 119905 isin [0 119879]
(9)
Next we give the definition of the Hamiltonian process
Definition 3 (see [19]) We define a Hamiltonian process
119867 [0 119879] timesR timesR timesR times 1198712(]) times 119880 timesR 997888rarr R (10)
as follows119867(119905 119886 119909 119910 119896 119906 120582)
= 119891 (119905 119886 119909 119910 119896 119906) + 120582 (119905) 119892 (119905 119886 119909 119910 119906)
+ 119887 (119905 119886 119906) 119901 (119905) + 120590 (119905 119886 119906) 119902 (119905)
+ int
R0
120574 (119905 119886 119906 119911) 119903 (119905 119911) ] (119889119911)
(11)
where 119867 is Frechet differentiable in the variables 119886 119909 119910 119896nabla119896119867 denotes the Frechet derivative in 119896 of 119867 the adjoint
processes 119901(119905) 119902(119905) 119903(119905 119911) and 120582(119905) are given by a pair offorward-backward SDEs as follows
(i) Forward system in the unknown process 120582(119905)
119889120582 (119905)
=
120597119867
120597119909
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot)
119906 (119905) 120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119905
+
120597119867
120597119910
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)
120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119861 (119905)
+ int
R0
nabla119896119867(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905)
120582 (119905) 119901 (119905) 119902 (119905) 119903 (119905 119911)) (119889119905 119889119911)
120582 (0) = ℎ1015840
1(119883 (0))
(12)
(ii) Backward system in the unknown processes119901(119905) 119902(119905)and 119903(119905 sdot)
119889119901 (119905)
= minus
120597119867
120597119886
(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 (119905) 120582 (119905)
119901 (119905) 119902 (119905) 119903 (119905 119911)) 119889119905
+ 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = 120583120582 (119879) + ℎ1015840
2(119860 (119879))
(13)
For the sake of simplicity we use the short hand notationin the following
120597119887
120597119886
(119905 119860 (119905) 119906 (119905) 120596) =
120597119887
120597119886
(119905)
120597119887
120597119906
(119905 119860 (119905) 119906 (119905) 120596) =
120597119887
120597119906
(119905)
(14)
and similarly for (120597120590120597119886)(119905) (120597120590120597119906)(119905) (120597120574120597119886)(119905) (120597120574120597119906)(119905) (120597119891120597119886)(119905) (120597119891120597119909)(119905) (120597119891120597119910)(119905) (120597119891120597119906)(119905) nabla
119896119891(119905 119911)
(120597119892120597119886)(119905) (120597119892120597119909)(119905) (120597119892120597119910)(119905) and (120597119892120597119906)(119905)
4 Abstract and Applied Analysis
Theorem 4 (maximum principle) Let (119906 120585) isin AG withcorresponding solutions 119860(119905) 119883(119905) 119884(119905) 119870(119905 119911) and 120582(119905) of(2) (12) and (13) Assume that for all (119906 120585) isin AG the followinggrowth conditions hold
119864[int
119879
0
1198832(119905) ((
120597119867
120597119910
(119905))
2
+ int
R0
1003817100381710038171003817nabla119896119867(119905 119911)
1003817100381710038171003817
2] (119889119911)) 119889119905]lt infin
119864 [int
119879
0
1205822(119905) (119884
2(119905) + int
R0
1198702(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1198602(119905) (119902
2(119905) + int
R0
1199032(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1199012(119905) ((120590
2(119905))
2
+ int
R0
1205742(119905 119911) ] (119889119911)) 119889119905] lt infin
(15)
Then the following are equivalent(1) (119906 120585) is a critical point forJ(119906 120585) in the sense that
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
forall119887119900119906119899119889119890119889(120573 120589) isin AG
(16)
(2) Consider
119864[
120597
120597119906
119867(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 120582 (119905))119906=119906(119905)
|G119905] = 0
(17)
for aa (119905 120596) isin [0 119879] times Ω and
sum
120591119894le119879
119864[119901 (120591119894) 119862 (120591119894) +
120597119897
120597120585
(120591119894) minus 120582 (120591
119894)119863 (120591
119894) | G
120591119894] = 0
(18)
Proof Define
(119905 120573 120589) =
119889
119889119910
119860 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119883 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119884 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 119911 120573 120589) =
119889
119889119910
119870 (119905 119911 119906 + 119910120573 120585 + 119910120589) |119910=0
(19)
Then we have
(0 120573 120589) =
119889
119889119910
119860 (0 119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
(119879 120573 120589) =
119889
119889119910
119860 (119879 119906 + 119910120573 120585 + 119910120589) |119910=0
=
1
120583
(119879 120573 120589)
119889 (119905 120573 120589) = [
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)] 119889119905
+ int
119905
0
[
120597120590
120597119886
(119904) (119905 120573 120589) +
120597120590
120597119906
(119904) 120573 (119904)] 119889119861 (119904)
+ int
R0
[
120597120574
120597119886
(119905) (119905 120573 120589)+
120597120574
120597119906
(119905) 120573 (119905)](119889119905 119889119911)
+ 119862 (119905) 119889120589 (119905)
119889(119905 120573 120589) = minus [
120597119892
120597119886
(119905) (119905 120573 120589) +
120597119892
120597119909
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) (119905 120573 120589) +
120597119892
120597119906
(119905) 120573 (119905)] 119889119905
+ (119905 120573 120589) 119889119861 (119905)
+ int
R0
(119905 119911 120573 120589) (119889119905 119889119911) + 119863 (119905) 119889120589 (119905)
(20)
Firstly we prove (1) rArr (2) Assume that (1) holds Thenwe have
0 =
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 119864[
[
int
119879
0
120597119891
120597119886
(119905) (119905 120573 120589) +
120597119891
120597119909
(119905) (119905 120573 120589)
+
120597119891
120597119910
(119905) (119905 120573 120589)
+ int
R0
nabla119896119891 (119905 119911) (119905 119911 120573 120589) ] (119889119911)
+
120597119891
120597119906
(119905) 120573 (119905) 119889119905 + ℎ1015840
1(119883 (0)) (0 120573 120589)
+ ℎ1015840
2(119860 (119879)) (119879 120573 120589) + sum
120591119894le119879
120597119897
120597120578
(120591119894) 120589119894]
]
(21)
By Ito formula we get
119864 [ℎ1015840
1(119883 (0)) (0 120573 120589)]
= 119864 [120582 (0) (0 120573 120589)]
Abstract and Applied Analysis 5
= 119864[120582 (119879) (119879 120573 120589) minus int
119879
0
(119905 120573 120589)
120597119867
120597119909
(119905) 119889119905
minus int
119879
0
120597119867
120597119910
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
120582 (119905) (
120597119892
120597119886
(119905) (119905 120573 120589)
+
120597119892
120597119909
(119905) (119905 120573 120589) +
120597119892
120597119910
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) 120573 (119905)) 119889119905
minus int
119879
0
int
R0
nabla119896119867(119905 119911) (119905 119911 120573 120589) ] (119889119911) 119889119905
minussum
119894ge1
120582 (120591119894)119863 (120591
119894) 120589119894]
(22)
where 0 le 120591119894le 119879 Now we consider
119864 [ℎ1015840
2(119860 (119879)) (119879 120573 120589)]
= 119864 [(119901 (119879) minus 120583120582 (119879)) (119879 120573 120589)]
= 119864 [119901 (119879) (119879 120573 120589)] minus 119864 [120582 (119879) (119879 120573 120589)]
(23)
Applying Ito formula to 119864[119901(119879)(119879 120573 120589)] we get
119864 [119901 (119879) (119879 120573 120589)]
= 119864 [119901 (0) (0 120573 120589)
+ int
119879
0
119901 (119905) (
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)) 119889119905
+sum
119894ge1
119901 (120591119894) 119862 (120591119894) 120589119894minus int
119879
0
120597119867
120597119886
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
(
120597120590
120597119886
(119905) (119905 120573 120589) +
120597120590
120597119906
(119905) 120573 (119905)) 119902 (119905) 119889119905
+ int
119879
0
int
R0
119903 (119905 119911) (
120597120574
120597119886
(119905 119911) (119905 120573 120589)
+
120597120574
120597119906
(119905 119911) 120573 (119905)) ] (119889119911) 119889119905]
(24)
where 0 le 120591119894le 119879 By substituting (22) (23) and (24) into
(21) we obtain
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
(
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
minus
120597119867
120597119886
(119905)) (119905 120573 120589)
+ (
120597119891
120597119909
(119905) + 120582 (119905)
120597119892
120597119909
(119905) minus
120597119867
120597119909
(119905)) (119905 120573 120589)
+ (
120597119891
120597119910
(119905) + 120582 (119905)
120597119892
120597119910
(119905) minus
120597119867
120597119910
(119905)) (119905 120573 120589)
+ int
R0
(nabla119896119891 (119905 119911) minus nabla
119896119867(119905 119911)) (119905 119911 120573 120589) ] (119889119911)
+ (
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905)
+int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)) 120573 (119905) 119889119905]
(25)
Depending on the definition of Hamiltonian119867 we get
120597119867
120597119909
(119905) =
120597119891
120597119909
(119905) +
120597119892
120597119909
(119905) 120582 (119905)
120597119867
120597119910
(119905) =
120597119891
120597119910
(119905) +
120597119892
120597119910
(119905) 120582 (119905)
nabla119896119867(119905 119911) = nabla
119896119891 (119905 119911)
120597119867
120597119886
(119905) =
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
120597119867
120597119906
(119905) =
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905) + int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)
(26)
6 Abstract and Applied Analysis
Hence (25) simplifies to
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905]
(27)
for all bounded (120573 120589) isin AG It is obvious that 120573(119905) isindependent of 120589(119905) 0 le 119905 le 119879 So we obtain from (27) that
119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905] = 0 (28)
119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894] = 0
(29)
holds for all bounded 120573 isin UG and 120589 isin IGNow we prove that (17) holds for all 120573(119905) isin UG We know
that (28) holds for all bounded 120573 isin UG So (28) holds for allbounded 120573 isin UG of the form
120573 (119905) = 120573119904(119905 120596) = 120579 (120596) 120594
[119904119879](119905) 119905 isin [0 119879] (30)
for a fixed 119904 isin [0 119879) where 120579(120596) is a boundedG119904-measurable
random variable Then we have
119864[
120597119867
120597119906
(119904) 120579] = 0 (31)
which holds for all boundedG119904-measurable random variable
120579 As a result we conclude that
119864[
120597119867
120597119906
(119904)
10038161003816100381610038161003816100381610038161003816
G119904] = 0 (32)
Moreover since (29) holds for all bounded G120591119894-measurable
random variable 120589119894 we conclude that (18) holds Therefore
we conclude that (1) rArr (2)(2) rArr (1) Each bounded 120573 isin U
119866can be approximated by
linear combinations of controls 120573119904of the formThenwe prove
that (2) rArr (1) by reversing the above argument
Remark 5 Let 119909 rarr ℎ1(119909) 119886 rarr ℎ
2(119886) 120585 rarr 119897(119905 120585) and
(119905 119886 119909 119896 119906) rarr 119867(119905 119886 119909 119910 119896 119906 120582) be concave for all 119905 isin
[0 119879]Then the local critical point (119906 120585) which is obtained byTheorem 4 is also a global optimum for the control problem(7)
Remark 6 Let G119905= F119905 119870(119905 119911) = 0 and 120574(119905 119886 119906 119911) = 0
Then ourmaximumprinciple (Theorem 4) coincideswith themaximum principle (Theorem 31) in [11]
4 Application
Example 7 (portfolio optimization problem) In a financialmarket we are given a subfiltration
G119905sube F119905
forall119905 isin [0 119879] (33)
representing the information available to the trader at time 119905Let 120585(119905) = sum
119894ge11205851198941[120591119894 119879]
(119905) 119905 le 119879 be a piecewiseconsumption process (see eg [11]) where 120591
119894 is a fixed
sequence of increasing G119905-stopping times and each 120585
119894is an
G120591119894-measurable random variable Then the wealth process
119860(119905) = 119860119906120578(119905) corresponding to the portfolio 119906(119905) is given
by
119889119860 (119905) = 119906 (119905) [120577 (119905) 119889119905 + 120587 (119905) 119889119861 (119905)
+int
R0
984858 (119905 119911) (119889119905 119889119911)] minus 120603119889120585 (119905)
119860 (0) = 119886 gt 0
(34)
where 120603 ge 0 R0= R 0 and 120587(119905) and 984858(119905 119911) are F
119905-
predictable processes such that 984858(119905 119911) ge minus1+120598 for some 120598 gt 0
and
int
119879
0
1003816100381610038161003816120577 (119905)
1003816100381610038161003816+ 1205872(119905) + int
R0
9848582(119905 119911) ] (119889119911) 119889119905 lt infin as
(35)
Endowed with initial wealth 119886 gt 0 an investor wantsto find a portfolio strategy 119906(sdot) and a consumption strategy120585(sdot) minimizing an expected functional which composes ofthree parts the first part is the total utility of the consumptionminusint
119879
0(1199062(119905)2)119889119905 the second part represents the risk of the
terminal wealth 120588(119860(119879)) = 119883minus119860119906(119879)
119892(0) where 119883minus119860119906(119879)
119892(0)
is the value at 119905 = 0 of the solution 119883(119905) of the followingbackward stochastic differential equation ([19])
119889119883 (119905) = minus119892 (119905 119883 (119905)) 119889119905 + 119884 (119905) 119889119861 (119905)
+ int
R0
119870 (119905 119911) (119889119905 119889119911) minus 120599119889120585 (119905)
119883 (119879) = minus119860 (119879)
(36)
and the third part is the utility derived from the consump-tion process 120585(sdot) More precisely for any admissible control(119906(sdot) 120585(sdot)) the utility functional is defined by
119869 (119906 (sdot) 120585 (sdot)) = 119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
(37)
where119864 denotes the expectation with respect to the probabil-ity measure 119875 and 119878 gt 0 Therefore the control problem is tofind Φ(119886) and (119906lowast(sdot) 120585lowast(sdot)) such that
Φ (119886) = inf(119906120585)isinAG
119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
= 119864[
[
minusint
119879
0
119906lowast2(119905)
2
119889119905 + 119883minus119860119906lowast (119879)
119892(0) +
119878
2
sum
120591119894le119879
120585lowast2
119894]
]
(38)
Abstract and Applied Analysis 7
The control problem (38) is a classical and impulsecontrols problem of forward-backward systems driven byLevy processes under partial information G
119905 Next we solve
the control problem (38) byTheorem 4 With the notation ofthe previous section we see that in Example 7 we have
119891 (119905 119886 119909 119910 119896 119906 120596) = minus
1199062
2
ℎ1(119909) = 119909 ℎ
2(119904 120596) = 0
119887 (119905 119886 119906 120596) = 119906120577 (119905) 120590 (119905 119886 119906 120596) = 119906120587 (119905)
120574 (119905 119886 119906 119911 120596) = 119906984858 (119905 119911) 119897 (120591119894 120585119894) =
119878
2
1205852
119894
119862 (119905) = 120603 119863 (119905) = 120599 120583 = minus1
(39)
Then by (11) the Hamiltonian is
119867(119905 119886 119909 119910 119896 119906 120582 120596) = minus
1199062
2
+ 120582 (119905) 119892 (119905 119909)
+ 119906 (119905) 120577 (119905) 119901 (119905) + 119906 (119905) 120587 (119905) 119902 (119905)
+ int
R0
119906 (119905) 984858 (119905 119911) 119903 (119905 119911) ] (119889119911)
(40)
where
119889119901 (119905) = 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = minus 120582 (119879)
(41)
and 120582(119905) is given by (12) that is
119889120582 (119905) = 120582 (119905) 119892119909(119905 119883 (119905)) 119889119905
120582 (0) = 1
(42)
where 119892119909(119905 119909) = (120597120597119909)119892(119905 119909) We can easily obtain the
solution of (42) as follows
120582 (119905) = expint119905
0
119892119909(119904 119883 (119904)) 119889119904 0 le 119905 le 119879 (43)
If (119906lowast(119905) 120585lowast(119905)) is a local critical point with corresponding119883lowast(119905) = 119883
(119906lowast)(119905) then by the sufficient and necessary
optimality condition (17) in Theorem 4 we get
119864 [119906lowast(119905) | G
119905] = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(44)
Since 119906lowast(119905) isG119905-adapted we have
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(45)
where 119901(119905) 119902(119905) and 119903(119905 119911) are given by (41)
On the other hand by the sufficient and necessary opti-mality condition (17) in Theorem 4 we obtain
sum
120591119894lt119879
119864 [119878120585lowast
119894+ 120603119901 (120591
119894) minus 120599120582 (120591
119894) | G120591119894] = 0 (46)
That is for each 120591119894lt 119879 we have
119864 [120585lowast
119894| G120591119894] =
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (47)
Since 120585119894is anG
120591119894-measurable random variable we have
120585lowast
119894=
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (48)
where 120582(119905) is given by (43) and 119901(119905) is given by (41) Con-sequently we summarize the above results in the followingtheorem
Theorem 8 Let 119901(119905) 119902(119905) and 119903(119905 119911) be the solutions of (41)and let 120582(119905) be the solution of (43) Then the pair (119906lowast(119905) 120585lowast(119905))is given by
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
120585lowast(119905) = sum
119894ge1
120585lowast
1198941[120591119894 119879]
(119905) 119905 le 119879
(49)
where 120585lowast119894given by (48) is the local critical point of the classical
and impulse controls problem (38)
5 Conclusion
We consider the partial information classical and impulsecontrols problem of forward-backward systems driven byLevy processes The control variable consists of two com-ponents the classical stochastic control and the impulsecontrol Because of the non-Markovian nature of the par-tial information dynamic programming principle cannotbe used to solve partial information control problems Asa result we derive a maximum principle for this partialinformation problem Because the concavity conditions of theutility functions and the Hamiltonian process may not holdin many applications we give the sufficient and necessaryoptimality conditions for the local critical points of thecontrol problem To illustrate the theoretical results we usethe maximum principle to solve a portfolio optimizationproblem with piecewise consumption processes and give itsexplicit solutions
In this paper we assume that the two different jumps inour system do not occur at the same time (Assumption 1)This assumption makes the problem easier to analyze How-ever it may fail in many applications Without this assump-tion it requiresmore attention to distinguish between the twodifferent jumpsThiswill be explored in our subsequentwork
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
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Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Theorem 4 (maximum principle) Let (119906 120585) isin AG withcorresponding solutions 119860(119905) 119883(119905) 119884(119905) 119870(119905 119911) and 120582(119905) of(2) (12) and (13) Assume that for all (119906 120585) isin AG the followinggrowth conditions hold
119864[int
119879
0
1198832(119905) ((
120597119867
120597119910
(119905))
2
+ int
R0
1003817100381710038171003817nabla119896119867(119905 119911)
1003817100381710038171003817
2] (119889119911)) 119889119905]lt infin
119864 [int
119879
0
1205822(119905) (119884
2(119905) + int
R0
1198702(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1198602(119905) (119902
2(119905) + int
R0
1199032(119905 119911) ] (119889119911)) 119889119905] lt infin
119864 [int
119879
0
1199012(119905) ((120590
2(119905))
2
+ int
R0
1205742(119905 119911) ] (119889119911)) 119889119905] lt infin
(15)
Then the following are equivalent(1) (119906 120585) is a critical point forJ(119906 120585) in the sense that
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
forall119887119900119906119899119889119890119889(120573 120589) isin AG
(16)
(2) Consider
119864[
120597
120597119906
119867(119905 119860 (119905) 119883 (119905) 119884 (119905) 119870 (119905 sdot) 119906 120582 (119905))119906=119906(119905)
|G119905] = 0
(17)
for aa (119905 120596) isin [0 119879] times Ω and
sum
120591119894le119879
119864[119901 (120591119894) 119862 (120591119894) +
120597119897
120597120585
(120591119894) minus 120582 (120591
119894)119863 (120591
119894) | G
120591119894] = 0
(18)
Proof Define
(119905 120573 120589) =
119889
119889119910
119860 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119883 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 120573 120589) =
119889
119889119910
119884 (119905 119906 + 119910120573 120585 + 119910120589) |119910=0
(119905 119911 120573 120589) =
119889
119889119910
119870 (119905 119911 119906 + 119910120573 120585 + 119910120589) |119910=0
(19)
Then we have
(0 120573 120589) =
119889
119889119910
119860 (0 119906 + 119910120573 120585 + 119910120589) |119910=0
= 0
(119879 120573 120589) =
119889
119889119910
119860 (119879 119906 + 119910120573 120585 + 119910120589) |119910=0
=
1
120583
(119879 120573 120589)
119889 (119905 120573 120589) = [
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)] 119889119905
+ int
119905
0
[
120597120590
120597119886
(119904) (119905 120573 120589) +
120597120590
120597119906
(119904) 120573 (119904)] 119889119861 (119904)
+ int
R0
[
120597120574
120597119886
(119905) (119905 120573 120589)+
120597120574
120597119906
(119905) 120573 (119905)](119889119905 119889119911)
+ 119862 (119905) 119889120589 (119905)
119889(119905 120573 120589) = minus [
120597119892
120597119886
(119905) (119905 120573 120589) +
120597119892
120597119909
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) (119905 120573 120589) +
120597119892
120597119906
(119905) 120573 (119905)] 119889119905
+ (119905 120573 120589) 119889119861 (119905)
+ int
R0
(119905 119911 120573 120589) (119889119905 119889119911) + 119863 (119905) 119889120589 (119905)
(20)
Firstly we prove (1) rArr (2) Assume that (1) holds Thenwe have
0 =
119889
119889119910
J (119906 + 119910120573 120585 + 119910120589) |119910=0
= 119864[
[
int
119879
0
120597119891
120597119886
(119905) (119905 120573 120589) +
120597119891
120597119909
(119905) (119905 120573 120589)
+
120597119891
120597119910
(119905) (119905 120573 120589)
+ int
R0
nabla119896119891 (119905 119911) (119905 119911 120573 120589) ] (119889119911)
+
120597119891
120597119906
(119905) 120573 (119905) 119889119905 + ℎ1015840
1(119883 (0)) (0 120573 120589)
+ ℎ1015840
2(119860 (119879)) (119879 120573 120589) + sum
120591119894le119879
120597119897
120597120578
(120591119894) 120589119894]
]
(21)
By Ito formula we get
119864 [ℎ1015840
1(119883 (0)) (0 120573 120589)]
= 119864 [120582 (0) (0 120573 120589)]
Abstract and Applied Analysis 5
= 119864[120582 (119879) (119879 120573 120589) minus int
119879
0
(119905 120573 120589)
120597119867
120597119909
(119905) 119889119905
minus int
119879
0
120597119867
120597119910
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
120582 (119905) (
120597119892
120597119886
(119905) (119905 120573 120589)
+
120597119892
120597119909
(119905) (119905 120573 120589) +
120597119892
120597119910
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) 120573 (119905)) 119889119905
minus int
119879
0
int
R0
nabla119896119867(119905 119911) (119905 119911 120573 120589) ] (119889119911) 119889119905
minussum
119894ge1
120582 (120591119894)119863 (120591
119894) 120589119894]
(22)
where 0 le 120591119894le 119879 Now we consider
119864 [ℎ1015840
2(119860 (119879)) (119879 120573 120589)]
= 119864 [(119901 (119879) minus 120583120582 (119879)) (119879 120573 120589)]
= 119864 [119901 (119879) (119879 120573 120589)] minus 119864 [120582 (119879) (119879 120573 120589)]
(23)
Applying Ito formula to 119864[119901(119879)(119879 120573 120589)] we get
119864 [119901 (119879) (119879 120573 120589)]
= 119864 [119901 (0) (0 120573 120589)
+ int
119879
0
119901 (119905) (
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)) 119889119905
+sum
119894ge1
119901 (120591119894) 119862 (120591119894) 120589119894minus int
119879
0
120597119867
120597119886
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
(
120597120590
120597119886
(119905) (119905 120573 120589) +
120597120590
120597119906
(119905) 120573 (119905)) 119902 (119905) 119889119905
+ int
119879
0
int
R0
119903 (119905 119911) (
120597120574
120597119886
(119905 119911) (119905 120573 120589)
+
120597120574
120597119906
(119905 119911) 120573 (119905)) ] (119889119911) 119889119905]
(24)
where 0 le 120591119894le 119879 By substituting (22) (23) and (24) into
(21) we obtain
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
(
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
minus
120597119867
120597119886
(119905)) (119905 120573 120589)
+ (
120597119891
120597119909
(119905) + 120582 (119905)
120597119892
120597119909
(119905) minus
120597119867
120597119909
(119905)) (119905 120573 120589)
+ (
120597119891
120597119910
(119905) + 120582 (119905)
120597119892
120597119910
(119905) minus
120597119867
120597119910
(119905)) (119905 120573 120589)
+ int
R0
(nabla119896119891 (119905 119911) minus nabla
119896119867(119905 119911)) (119905 119911 120573 120589) ] (119889119911)
+ (
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905)
+int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)) 120573 (119905) 119889119905]
(25)
Depending on the definition of Hamiltonian119867 we get
120597119867
120597119909
(119905) =
120597119891
120597119909
(119905) +
120597119892
120597119909
(119905) 120582 (119905)
120597119867
120597119910
(119905) =
120597119891
120597119910
(119905) +
120597119892
120597119910
(119905) 120582 (119905)
nabla119896119867(119905 119911) = nabla
119896119891 (119905 119911)
120597119867
120597119886
(119905) =
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
120597119867
120597119906
(119905) =
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905) + int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)
(26)
6 Abstract and Applied Analysis
Hence (25) simplifies to
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905]
(27)
for all bounded (120573 120589) isin AG It is obvious that 120573(119905) isindependent of 120589(119905) 0 le 119905 le 119879 So we obtain from (27) that
119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905] = 0 (28)
119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894] = 0
(29)
holds for all bounded 120573 isin UG and 120589 isin IGNow we prove that (17) holds for all 120573(119905) isin UG We know
that (28) holds for all bounded 120573 isin UG So (28) holds for allbounded 120573 isin UG of the form
120573 (119905) = 120573119904(119905 120596) = 120579 (120596) 120594
[119904119879](119905) 119905 isin [0 119879] (30)
for a fixed 119904 isin [0 119879) where 120579(120596) is a boundedG119904-measurable
random variable Then we have
119864[
120597119867
120597119906
(119904) 120579] = 0 (31)
which holds for all boundedG119904-measurable random variable
120579 As a result we conclude that
119864[
120597119867
120597119906
(119904)
10038161003816100381610038161003816100381610038161003816
G119904] = 0 (32)
Moreover since (29) holds for all bounded G120591119894-measurable
random variable 120589119894 we conclude that (18) holds Therefore
we conclude that (1) rArr (2)(2) rArr (1) Each bounded 120573 isin U
119866can be approximated by
linear combinations of controls 120573119904of the formThenwe prove
that (2) rArr (1) by reversing the above argument
Remark 5 Let 119909 rarr ℎ1(119909) 119886 rarr ℎ
2(119886) 120585 rarr 119897(119905 120585) and
(119905 119886 119909 119896 119906) rarr 119867(119905 119886 119909 119910 119896 119906 120582) be concave for all 119905 isin
[0 119879]Then the local critical point (119906 120585) which is obtained byTheorem 4 is also a global optimum for the control problem(7)
Remark 6 Let G119905= F119905 119870(119905 119911) = 0 and 120574(119905 119886 119906 119911) = 0
Then ourmaximumprinciple (Theorem 4) coincideswith themaximum principle (Theorem 31) in [11]
4 Application
Example 7 (portfolio optimization problem) In a financialmarket we are given a subfiltration
G119905sube F119905
forall119905 isin [0 119879] (33)
representing the information available to the trader at time 119905Let 120585(119905) = sum
119894ge11205851198941[120591119894 119879]
(119905) 119905 le 119879 be a piecewiseconsumption process (see eg [11]) where 120591
119894 is a fixed
sequence of increasing G119905-stopping times and each 120585
119894is an
G120591119894-measurable random variable Then the wealth process
119860(119905) = 119860119906120578(119905) corresponding to the portfolio 119906(119905) is given
by
119889119860 (119905) = 119906 (119905) [120577 (119905) 119889119905 + 120587 (119905) 119889119861 (119905)
+int
R0
984858 (119905 119911) (119889119905 119889119911)] minus 120603119889120585 (119905)
119860 (0) = 119886 gt 0
(34)
where 120603 ge 0 R0= R 0 and 120587(119905) and 984858(119905 119911) are F
119905-
predictable processes such that 984858(119905 119911) ge minus1+120598 for some 120598 gt 0
and
int
119879
0
1003816100381610038161003816120577 (119905)
1003816100381610038161003816+ 1205872(119905) + int
R0
9848582(119905 119911) ] (119889119911) 119889119905 lt infin as
(35)
Endowed with initial wealth 119886 gt 0 an investor wantsto find a portfolio strategy 119906(sdot) and a consumption strategy120585(sdot) minimizing an expected functional which composes ofthree parts the first part is the total utility of the consumptionminusint
119879
0(1199062(119905)2)119889119905 the second part represents the risk of the
terminal wealth 120588(119860(119879)) = 119883minus119860119906(119879)
119892(0) where 119883minus119860119906(119879)
119892(0)
is the value at 119905 = 0 of the solution 119883(119905) of the followingbackward stochastic differential equation ([19])
119889119883 (119905) = minus119892 (119905 119883 (119905)) 119889119905 + 119884 (119905) 119889119861 (119905)
+ int
R0
119870 (119905 119911) (119889119905 119889119911) minus 120599119889120585 (119905)
119883 (119879) = minus119860 (119879)
(36)
and the third part is the utility derived from the consump-tion process 120585(sdot) More precisely for any admissible control(119906(sdot) 120585(sdot)) the utility functional is defined by
119869 (119906 (sdot) 120585 (sdot)) = 119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
(37)
where119864 denotes the expectation with respect to the probabil-ity measure 119875 and 119878 gt 0 Therefore the control problem is tofind Φ(119886) and (119906lowast(sdot) 120585lowast(sdot)) such that
Φ (119886) = inf(119906120585)isinAG
119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
= 119864[
[
minusint
119879
0
119906lowast2(119905)
2
119889119905 + 119883minus119860119906lowast (119879)
119892(0) +
119878
2
sum
120591119894le119879
120585lowast2
119894]
]
(38)
Abstract and Applied Analysis 7
The control problem (38) is a classical and impulsecontrols problem of forward-backward systems driven byLevy processes under partial information G
119905 Next we solve
the control problem (38) byTheorem 4 With the notation ofthe previous section we see that in Example 7 we have
119891 (119905 119886 119909 119910 119896 119906 120596) = minus
1199062
2
ℎ1(119909) = 119909 ℎ
2(119904 120596) = 0
119887 (119905 119886 119906 120596) = 119906120577 (119905) 120590 (119905 119886 119906 120596) = 119906120587 (119905)
120574 (119905 119886 119906 119911 120596) = 119906984858 (119905 119911) 119897 (120591119894 120585119894) =
119878
2
1205852
119894
119862 (119905) = 120603 119863 (119905) = 120599 120583 = minus1
(39)
Then by (11) the Hamiltonian is
119867(119905 119886 119909 119910 119896 119906 120582 120596) = minus
1199062
2
+ 120582 (119905) 119892 (119905 119909)
+ 119906 (119905) 120577 (119905) 119901 (119905) + 119906 (119905) 120587 (119905) 119902 (119905)
+ int
R0
119906 (119905) 984858 (119905 119911) 119903 (119905 119911) ] (119889119911)
(40)
where
119889119901 (119905) = 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = minus 120582 (119879)
(41)
and 120582(119905) is given by (12) that is
119889120582 (119905) = 120582 (119905) 119892119909(119905 119883 (119905)) 119889119905
120582 (0) = 1
(42)
where 119892119909(119905 119909) = (120597120597119909)119892(119905 119909) We can easily obtain the
solution of (42) as follows
120582 (119905) = expint119905
0
119892119909(119904 119883 (119904)) 119889119904 0 le 119905 le 119879 (43)
If (119906lowast(119905) 120585lowast(119905)) is a local critical point with corresponding119883lowast(119905) = 119883
(119906lowast)(119905) then by the sufficient and necessary
optimality condition (17) in Theorem 4 we get
119864 [119906lowast(119905) | G
119905] = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(44)
Since 119906lowast(119905) isG119905-adapted we have
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(45)
where 119901(119905) 119902(119905) and 119903(119905 119911) are given by (41)
On the other hand by the sufficient and necessary opti-mality condition (17) in Theorem 4 we obtain
sum
120591119894lt119879
119864 [119878120585lowast
119894+ 120603119901 (120591
119894) minus 120599120582 (120591
119894) | G120591119894] = 0 (46)
That is for each 120591119894lt 119879 we have
119864 [120585lowast
119894| G120591119894] =
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (47)
Since 120585119894is anG
120591119894-measurable random variable we have
120585lowast
119894=
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (48)
where 120582(119905) is given by (43) and 119901(119905) is given by (41) Con-sequently we summarize the above results in the followingtheorem
Theorem 8 Let 119901(119905) 119902(119905) and 119903(119905 119911) be the solutions of (41)and let 120582(119905) be the solution of (43) Then the pair (119906lowast(119905) 120585lowast(119905))is given by
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
120585lowast(119905) = sum
119894ge1
120585lowast
1198941[120591119894 119879]
(119905) 119905 le 119879
(49)
where 120585lowast119894given by (48) is the local critical point of the classical
and impulse controls problem (38)
5 Conclusion
We consider the partial information classical and impulsecontrols problem of forward-backward systems driven byLevy processes The control variable consists of two com-ponents the classical stochastic control and the impulsecontrol Because of the non-Markovian nature of the par-tial information dynamic programming principle cannotbe used to solve partial information control problems Asa result we derive a maximum principle for this partialinformation problem Because the concavity conditions of theutility functions and the Hamiltonian process may not holdin many applications we give the sufficient and necessaryoptimality conditions for the local critical points of thecontrol problem To illustrate the theoretical results we usethe maximum principle to solve a portfolio optimizationproblem with piecewise consumption processes and give itsexplicit solutions
In this paper we assume that the two different jumps inour system do not occur at the same time (Assumption 1)This assumption makes the problem easier to analyze How-ever it may fail in many applications Without this assump-tion it requiresmore attention to distinguish between the twodifferent jumpsThiswill be explored in our subsequentwork
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
= 119864[120582 (119879) (119879 120573 120589) minus int
119879
0
(119905 120573 120589)
120597119867
120597119909
(119905) 119889119905
minus int
119879
0
120597119867
120597119910
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
120582 (119905) (
120597119892
120597119886
(119905) (119905 120573 120589)
+
120597119892
120597119909
(119905) (119905 120573 120589) +
120597119892
120597119910
(119905) (119905 120573 120589)
+
120597119892
120597119910
(119905) 120573 (119905)) 119889119905
minus int
119879
0
int
R0
nabla119896119867(119905 119911) (119905 119911 120573 120589) ] (119889119911) 119889119905
minussum
119894ge1
120582 (120591119894)119863 (120591
119894) 120589119894]
(22)
where 0 le 120591119894le 119879 Now we consider
119864 [ℎ1015840
2(119860 (119879)) (119879 120573 120589)]
= 119864 [(119901 (119879) minus 120583120582 (119879)) (119879 120573 120589)]
= 119864 [119901 (119879) (119879 120573 120589)] minus 119864 [120582 (119879) (119879 120573 120589)]
(23)
Applying Ito formula to 119864[119901(119879)(119879 120573 120589)] we get
119864 [119901 (119879) (119879 120573 120589)]
= 119864 [119901 (0) (0 120573 120589)
+ int
119879
0
119901 (119905) (
120597119887
120597119886
(119905) (119905 120573 120589) +
120597119887
120597119906
(119905) 120573 (119905)) 119889119905
+sum
119894ge1
119901 (120591119894) 119862 (120591119894) 120589119894minus int
119879
0
120597119867
120597119886
(119905) (119905 120573 120589) 119889119905
+ int
119879
0
(
120597120590
120597119886
(119905) (119905 120573 120589) +
120597120590
120597119906
(119905) 120573 (119905)) 119902 (119905) 119889119905
+ int
119879
0
int
R0
119903 (119905 119911) (
120597120574
120597119886
(119905 119911) (119905 120573 120589)
+
120597120574
120597119906
(119905 119911) 120573 (119905)) ] (119889119911) 119889119905]
(24)
where 0 le 120591119894le 119879 By substituting (22) (23) and (24) into
(21) we obtain
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
(
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
minus
120597119867
120597119886
(119905)) (119905 120573 120589)
+ (
120597119891
120597119909
(119905) + 120582 (119905)
120597119892
120597119909
(119905) minus
120597119867
120597119909
(119905)) (119905 120573 120589)
+ (
120597119891
120597119910
(119905) + 120582 (119905)
120597119892
120597119910
(119905) minus
120597119867
120597119910
(119905)) (119905 120573 120589)
+ int
R0
(nabla119896119891 (119905 119911) minus nabla
119896119867(119905 119911)) (119905 119911 120573 120589) ] (119889119911)
+ (
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905)
+int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)) 120573 (119905) 119889119905]
(25)
Depending on the definition of Hamiltonian119867 we get
120597119867
120597119909
(119905) =
120597119891
120597119909
(119905) +
120597119892
120597119909
(119905) 120582 (119905)
120597119867
120597119910
(119905) =
120597119891
120597119910
(119905) +
120597119892
120597119910
(119905) 120582 (119905)
nabla119896119867(119905 119911) = nabla
119896119891 (119905 119911)
120597119867
120597119886
(119905) =
120597119891
120597119886
(119905) + 120582 (119905)
120597119892
120597119886
(119905) + 119901 (119905)
120597119887
120597119886
(119905)
+ 119902 (119905)
120597120590
120597119886
(119905) + int
R0
120597120574
120597119886
(119905 119911) 119903 (119905 119911) ] (119889119911)
120597119867
120597119906
(119905) =
120597119891
120597119906
(119905) + 120582 (119905)
120597119892
120597119906
(119905) + 119901 (119905)
120597119887
120597119906
(119905)
+ 119902 (119905)
120597120590
120597119906
(119905) + int
R0
119903 (119905 119911)
120597120574
120597119906
(119905 119911) ] (119889119911)
(26)
6 Abstract and Applied Analysis
Hence (25) simplifies to
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905]
(27)
for all bounded (120573 120589) isin AG It is obvious that 120573(119905) isindependent of 120589(119905) 0 le 119905 le 119879 So we obtain from (27) that
119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905] = 0 (28)
119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894] = 0
(29)
holds for all bounded 120573 isin UG and 120589 isin IGNow we prove that (17) holds for all 120573(119905) isin UG We know
that (28) holds for all bounded 120573 isin UG So (28) holds for allbounded 120573 isin UG of the form
120573 (119905) = 120573119904(119905 120596) = 120579 (120596) 120594
[119904119879](119905) 119905 isin [0 119879] (30)
for a fixed 119904 isin [0 119879) where 120579(120596) is a boundedG119904-measurable
random variable Then we have
119864[
120597119867
120597119906
(119904) 120579] = 0 (31)
which holds for all boundedG119904-measurable random variable
120579 As a result we conclude that
119864[
120597119867
120597119906
(119904)
10038161003816100381610038161003816100381610038161003816
G119904] = 0 (32)
Moreover since (29) holds for all bounded G120591119894-measurable
random variable 120589119894 we conclude that (18) holds Therefore
we conclude that (1) rArr (2)(2) rArr (1) Each bounded 120573 isin U
119866can be approximated by
linear combinations of controls 120573119904of the formThenwe prove
that (2) rArr (1) by reversing the above argument
Remark 5 Let 119909 rarr ℎ1(119909) 119886 rarr ℎ
2(119886) 120585 rarr 119897(119905 120585) and
(119905 119886 119909 119896 119906) rarr 119867(119905 119886 119909 119910 119896 119906 120582) be concave for all 119905 isin
[0 119879]Then the local critical point (119906 120585) which is obtained byTheorem 4 is also a global optimum for the control problem(7)
Remark 6 Let G119905= F119905 119870(119905 119911) = 0 and 120574(119905 119886 119906 119911) = 0
Then ourmaximumprinciple (Theorem 4) coincideswith themaximum principle (Theorem 31) in [11]
4 Application
Example 7 (portfolio optimization problem) In a financialmarket we are given a subfiltration
G119905sube F119905
forall119905 isin [0 119879] (33)
representing the information available to the trader at time 119905Let 120585(119905) = sum
119894ge11205851198941[120591119894 119879]
(119905) 119905 le 119879 be a piecewiseconsumption process (see eg [11]) where 120591
119894 is a fixed
sequence of increasing G119905-stopping times and each 120585
119894is an
G120591119894-measurable random variable Then the wealth process
119860(119905) = 119860119906120578(119905) corresponding to the portfolio 119906(119905) is given
by
119889119860 (119905) = 119906 (119905) [120577 (119905) 119889119905 + 120587 (119905) 119889119861 (119905)
+int
R0
984858 (119905 119911) (119889119905 119889119911)] minus 120603119889120585 (119905)
119860 (0) = 119886 gt 0
(34)
where 120603 ge 0 R0= R 0 and 120587(119905) and 984858(119905 119911) are F
119905-
predictable processes such that 984858(119905 119911) ge minus1+120598 for some 120598 gt 0
and
int
119879
0
1003816100381610038161003816120577 (119905)
1003816100381610038161003816+ 1205872(119905) + int
R0
9848582(119905 119911) ] (119889119911) 119889119905 lt infin as
(35)
Endowed with initial wealth 119886 gt 0 an investor wantsto find a portfolio strategy 119906(sdot) and a consumption strategy120585(sdot) minimizing an expected functional which composes ofthree parts the first part is the total utility of the consumptionminusint
119879
0(1199062(119905)2)119889119905 the second part represents the risk of the
terminal wealth 120588(119860(119879)) = 119883minus119860119906(119879)
119892(0) where 119883minus119860119906(119879)
119892(0)
is the value at 119905 = 0 of the solution 119883(119905) of the followingbackward stochastic differential equation ([19])
119889119883 (119905) = minus119892 (119905 119883 (119905)) 119889119905 + 119884 (119905) 119889119861 (119905)
+ int
R0
119870 (119905 119911) (119889119905 119889119911) minus 120599119889120585 (119905)
119883 (119879) = minus119860 (119879)
(36)
and the third part is the utility derived from the consump-tion process 120585(sdot) More precisely for any admissible control(119906(sdot) 120585(sdot)) the utility functional is defined by
119869 (119906 (sdot) 120585 (sdot)) = 119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
(37)
where119864 denotes the expectation with respect to the probabil-ity measure 119875 and 119878 gt 0 Therefore the control problem is tofind Φ(119886) and (119906lowast(sdot) 120585lowast(sdot)) such that
Φ (119886) = inf(119906120585)isinAG
119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
= 119864[
[
minusint
119879
0
119906lowast2(119905)
2
119889119905 + 119883minus119860119906lowast (119879)
119892(0) +
119878
2
sum
120591119894le119879
120585lowast2
119894]
]
(38)
Abstract and Applied Analysis 7
The control problem (38) is a classical and impulsecontrols problem of forward-backward systems driven byLevy processes under partial information G
119905 Next we solve
the control problem (38) byTheorem 4 With the notation ofthe previous section we see that in Example 7 we have
119891 (119905 119886 119909 119910 119896 119906 120596) = minus
1199062
2
ℎ1(119909) = 119909 ℎ
2(119904 120596) = 0
119887 (119905 119886 119906 120596) = 119906120577 (119905) 120590 (119905 119886 119906 120596) = 119906120587 (119905)
120574 (119905 119886 119906 119911 120596) = 119906984858 (119905 119911) 119897 (120591119894 120585119894) =
119878
2
1205852
119894
119862 (119905) = 120603 119863 (119905) = 120599 120583 = minus1
(39)
Then by (11) the Hamiltonian is
119867(119905 119886 119909 119910 119896 119906 120582 120596) = minus
1199062
2
+ 120582 (119905) 119892 (119905 119909)
+ 119906 (119905) 120577 (119905) 119901 (119905) + 119906 (119905) 120587 (119905) 119902 (119905)
+ int
R0
119906 (119905) 984858 (119905 119911) 119903 (119905 119911) ] (119889119911)
(40)
where
119889119901 (119905) = 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = minus 120582 (119879)
(41)
and 120582(119905) is given by (12) that is
119889120582 (119905) = 120582 (119905) 119892119909(119905 119883 (119905)) 119889119905
120582 (0) = 1
(42)
where 119892119909(119905 119909) = (120597120597119909)119892(119905 119909) We can easily obtain the
solution of (42) as follows
120582 (119905) = expint119905
0
119892119909(119904 119883 (119904)) 119889119904 0 le 119905 le 119879 (43)
If (119906lowast(119905) 120585lowast(119905)) is a local critical point with corresponding119883lowast(119905) = 119883
(119906lowast)(119905) then by the sufficient and necessary
optimality condition (17) in Theorem 4 we get
119864 [119906lowast(119905) | G
119905] = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(44)
Since 119906lowast(119905) isG119905-adapted we have
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(45)
where 119901(119905) 119902(119905) and 119903(119905 119911) are given by (41)
On the other hand by the sufficient and necessary opti-mality condition (17) in Theorem 4 we obtain
sum
120591119894lt119879
119864 [119878120585lowast
119894+ 120603119901 (120591
119894) minus 120599120582 (120591
119894) | G120591119894] = 0 (46)
That is for each 120591119894lt 119879 we have
119864 [120585lowast
119894| G120591119894] =
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (47)
Since 120585119894is anG
120591119894-measurable random variable we have
120585lowast
119894=
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (48)
where 120582(119905) is given by (43) and 119901(119905) is given by (41) Con-sequently we summarize the above results in the followingtheorem
Theorem 8 Let 119901(119905) 119902(119905) and 119903(119905 119911) be the solutions of (41)and let 120582(119905) be the solution of (43) Then the pair (119906lowast(119905) 120585lowast(119905))is given by
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
120585lowast(119905) = sum
119894ge1
120585lowast
1198941[120591119894 119879]
(119905) 119905 le 119879
(49)
where 120585lowast119894given by (48) is the local critical point of the classical
and impulse controls problem (38)
5 Conclusion
We consider the partial information classical and impulsecontrols problem of forward-backward systems driven byLevy processes The control variable consists of two com-ponents the classical stochastic control and the impulsecontrol Because of the non-Markovian nature of the par-tial information dynamic programming principle cannotbe used to solve partial information control problems Asa result we derive a maximum principle for this partialinformation problem Because the concavity conditions of theutility functions and the Hamiltonian process may not holdin many applications we give the sufficient and necessaryoptimality conditions for the local critical points of thecontrol problem To illustrate the theoretical results we usethe maximum principle to solve a portfolio optimizationproblem with piecewise consumption processes and give itsexplicit solutions
In this paper we assume that the two different jumps inour system do not occur at the same time (Assumption 1)This assumption makes the problem easier to analyze How-ever it may fail in many applications Without this assump-tion it requiresmore attention to distinguish between the twodifferent jumpsThiswill be explored in our subsequentwork
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Hence (25) simplifies to
0 = 119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894]
+ 119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905]
(27)
for all bounded (120573 120589) isin AG It is obvious that 120573(119905) isindependent of 120589(119905) 0 le 119905 le 119879 So we obtain from (27) that
119864[int
119879
0
120597119867
120597119906
(119905) 120573 (119905) 119889119905] = 0 (28)
119864[sum
119894ge1
(119901 (120591119894) 119862 (120591119894) +
120597119897
120597120578
(120591119894) minus 120582 (120591
119894)119863 (120591
119894)) 120589119894] = 0
(29)
holds for all bounded 120573 isin UG and 120589 isin IGNow we prove that (17) holds for all 120573(119905) isin UG We know
that (28) holds for all bounded 120573 isin UG So (28) holds for allbounded 120573 isin UG of the form
120573 (119905) = 120573119904(119905 120596) = 120579 (120596) 120594
[119904119879](119905) 119905 isin [0 119879] (30)
for a fixed 119904 isin [0 119879) where 120579(120596) is a boundedG119904-measurable
random variable Then we have
119864[
120597119867
120597119906
(119904) 120579] = 0 (31)
which holds for all boundedG119904-measurable random variable
120579 As a result we conclude that
119864[
120597119867
120597119906
(119904)
10038161003816100381610038161003816100381610038161003816
G119904] = 0 (32)
Moreover since (29) holds for all bounded G120591119894-measurable
random variable 120589119894 we conclude that (18) holds Therefore
we conclude that (1) rArr (2)(2) rArr (1) Each bounded 120573 isin U
119866can be approximated by
linear combinations of controls 120573119904of the formThenwe prove
that (2) rArr (1) by reversing the above argument
Remark 5 Let 119909 rarr ℎ1(119909) 119886 rarr ℎ
2(119886) 120585 rarr 119897(119905 120585) and
(119905 119886 119909 119896 119906) rarr 119867(119905 119886 119909 119910 119896 119906 120582) be concave for all 119905 isin
[0 119879]Then the local critical point (119906 120585) which is obtained byTheorem 4 is also a global optimum for the control problem(7)
Remark 6 Let G119905= F119905 119870(119905 119911) = 0 and 120574(119905 119886 119906 119911) = 0
Then ourmaximumprinciple (Theorem 4) coincideswith themaximum principle (Theorem 31) in [11]
4 Application
Example 7 (portfolio optimization problem) In a financialmarket we are given a subfiltration
G119905sube F119905
forall119905 isin [0 119879] (33)
representing the information available to the trader at time 119905Let 120585(119905) = sum
119894ge11205851198941[120591119894 119879]
(119905) 119905 le 119879 be a piecewiseconsumption process (see eg [11]) where 120591
119894 is a fixed
sequence of increasing G119905-stopping times and each 120585
119894is an
G120591119894-measurable random variable Then the wealth process
119860(119905) = 119860119906120578(119905) corresponding to the portfolio 119906(119905) is given
by
119889119860 (119905) = 119906 (119905) [120577 (119905) 119889119905 + 120587 (119905) 119889119861 (119905)
+int
R0
984858 (119905 119911) (119889119905 119889119911)] minus 120603119889120585 (119905)
119860 (0) = 119886 gt 0
(34)
where 120603 ge 0 R0= R 0 and 120587(119905) and 984858(119905 119911) are F
119905-
predictable processes such that 984858(119905 119911) ge minus1+120598 for some 120598 gt 0
and
int
119879
0
1003816100381610038161003816120577 (119905)
1003816100381610038161003816+ 1205872(119905) + int
R0
9848582(119905 119911) ] (119889119911) 119889119905 lt infin as
(35)
Endowed with initial wealth 119886 gt 0 an investor wantsto find a portfolio strategy 119906(sdot) and a consumption strategy120585(sdot) minimizing an expected functional which composes ofthree parts the first part is the total utility of the consumptionminusint
119879
0(1199062(119905)2)119889119905 the second part represents the risk of the
terminal wealth 120588(119860(119879)) = 119883minus119860119906(119879)
119892(0) where 119883minus119860119906(119879)
119892(0)
is the value at 119905 = 0 of the solution 119883(119905) of the followingbackward stochastic differential equation ([19])
119889119883 (119905) = minus119892 (119905 119883 (119905)) 119889119905 + 119884 (119905) 119889119861 (119905)
+ int
R0
119870 (119905 119911) (119889119905 119889119911) minus 120599119889120585 (119905)
119883 (119879) = minus119860 (119879)
(36)
and the third part is the utility derived from the consump-tion process 120585(sdot) More precisely for any admissible control(119906(sdot) 120585(sdot)) the utility functional is defined by
119869 (119906 (sdot) 120585 (sdot)) = 119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
(37)
where119864 denotes the expectation with respect to the probabil-ity measure 119875 and 119878 gt 0 Therefore the control problem is tofind Φ(119886) and (119906lowast(sdot) 120585lowast(sdot)) such that
Φ (119886) = inf(119906120585)isinAG
119864[
[
minusint
119879
0
1199062(119905)
2
119889119905 + 120588 (119860 (119879)) +
119878
2
sum
120591119894le119879
1205852
119894]
]
= 119864[
[
minusint
119879
0
119906lowast2(119905)
2
119889119905 + 119883minus119860119906lowast (119879)
119892(0) +
119878
2
sum
120591119894le119879
120585lowast2
119894]
]
(38)
Abstract and Applied Analysis 7
The control problem (38) is a classical and impulsecontrols problem of forward-backward systems driven byLevy processes under partial information G
119905 Next we solve
the control problem (38) byTheorem 4 With the notation ofthe previous section we see that in Example 7 we have
119891 (119905 119886 119909 119910 119896 119906 120596) = minus
1199062
2
ℎ1(119909) = 119909 ℎ
2(119904 120596) = 0
119887 (119905 119886 119906 120596) = 119906120577 (119905) 120590 (119905 119886 119906 120596) = 119906120587 (119905)
120574 (119905 119886 119906 119911 120596) = 119906984858 (119905 119911) 119897 (120591119894 120585119894) =
119878
2
1205852
119894
119862 (119905) = 120603 119863 (119905) = 120599 120583 = minus1
(39)
Then by (11) the Hamiltonian is
119867(119905 119886 119909 119910 119896 119906 120582 120596) = minus
1199062
2
+ 120582 (119905) 119892 (119905 119909)
+ 119906 (119905) 120577 (119905) 119901 (119905) + 119906 (119905) 120587 (119905) 119902 (119905)
+ int
R0
119906 (119905) 984858 (119905 119911) 119903 (119905 119911) ] (119889119911)
(40)
where
119889119901 (119905) = 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = minus 120582 (119879)
(41)
and 120582(119905) is given by (12) that is
119889120582 (119905) = 120582 (119905) 119892119909(119905 119883 (119905)) 119889119905
120582 (0) = 1
(42)
where 119892119909(119905 119909) = (120597120597119909)119892(119905 119909) We can easily obtain the
solution of (42) as follows
120582 (119905) = expint119905
0
119892119909(119904 119883 (119904)) 119889119904 0 le 119905 le 119879 (43)
If (119906lowast(119905) 120585lowast(119905)) is a local critical point with corresponding119883lowast(119905) = 119883
(119906lowast)(119905) then by the sufficient and necessary
optimality condition (17) in Theorem 4 we get
119864 [119906lowast(119905) | G
119905] = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(44)
Since 119906lowast(119905) isG119905-adapted we have
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(45)
where 119901(119905) 119902(119905) and 119903(119905 119911) are given by (41)
On the other hand by the sufficient and necessary opti-mality condition (17) in Theorem 4 we obtain
sum
120591119894lt119879
119864 [119878120585lowast
119894+ 120603119901 (120591
119894) minus 120599120582 (120591
119894) | G120591119894] = 0 (46)
That is for each 120591119894lt 119879 we have
119864 [120585lowast
119894| G120591119894] =
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (47)
Since 120585119894is anG
120591119894-measurable random variable we have
120585lowast
119894=
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (48)
where 120582(119905) is given by (43) and 119901(119905) is given by (41) Con-sequently we summarize the above results in the followingtheorem
Theorem 8 Let 119901(119905) 119902(119905) and 119903(119905 119911) be the solutions of (41)and let 120582(119905) be the solution of (43) Then the pair (119906lowast(119905) 120585lowast(119905))is given by
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
120585lowast(119905) = sum
119894ge1
120585lowast
1198941[120591119894 119879]
(119905) 119905 le 119879
(49)
where 120585lowast119894given by (48) is the local critical point of the classical
and impulse controls problem (38)
5 Conclusion
We consider the partial information classical and impulsecontrols problem of forward-backward systems driven byLevy processes The control variable consists of two com-ponents the classical stochastic control and the impulsecontrol Because of the non-Markovian nature of the par-tial information dynamic programming principle cannotbe used to solve partial information control problems Asa result we derive a maximum principle for this partialinformation problem Because the concavity conditions of theutility functions and the Hamiltonian process may not holdin many applications we give the sufficient and necessaryoptimality conditions for the local critical points of thecontrol problem To illustrate the theoretical results we usethe maximum principle to solve a portfolio optimizationproblem with piecewise consumption processes and give itsexplicit solutions
In this paper we assume that the two different jumps inour system do not occur at the same time (Assumption 1)This assumption makes the problem easier to analyze How-ever it may fail in many applications Without this assump-tion it requiresmore attention to distinguish between the twodifferent jumpsThiswill be explored in our subsequentwork
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
The control problem (38) is a classical and impulsecontrols problem of forward-backward systems driven byLevy processes under partial information G
119905 Next we solve
the control problem (38) byTheorem 4 With the notation ofthe previous section we see that in Example 7 we have
119891 (119905 119886 119909 119910 119896 119906 120596) = minus
1199062
2
ℎ1(119909) = 119909 ℎ
2(119904 120596) = 0
119887 (119905 119886 119906 120596) = 119906120577 (119905) 120590 (119905 119886 119906 120596) = 119906120587 (119905)
120574 (119905 119886 119906 119911 120596) = 119906984858 (119905 119911) 119897 (120591119894 120585119894) =
119878
2
1205852
119894
119862 (119905) = 120603 119863 (119905) = 120599 120583 = minus1
(39)
Then by (11) the Hamiltonian is
119867(119905 119886 119909 119910 119896 119906 120582 120596) = minus
1199062
2
+ 120582 (119905) 119892 (119905 119909)
+ 119906 (119905) 120577 (119905) 119901 (119905) + 119906 (119905) 120587 (119905) 119902 (119905)
+ int
R0
119906 (119905) 984858 (119905 119911) 119903 (119905 119911) ] (119889119911)
(40)
where
119889119901 (119905) = 119902 (119905) 119889119861 (119905) + int
R0
119903 (119905 119911) (119889119905 119889119911)
119901 (119879) = minus 120582 (119879)
(41)
and 120582(119905) is given by (12) that is
119889120582 (119905) = 120582 (119905) 119892119909(119905 119883 (119905)) 119889119905
120582 (0) = 1
(42)
where 119892119909(119905 119909) = (120597120597119909)119892(119905 119909) We can easily obtain the
solution of (42) as follows
120582 (119905) = expint119905
0
119892119909(119904 119883 (119904)) 119889119904 0 le 119905 le 119879 (43)
If (119906lowast(119905) 120585lowast(119905)) is a local critical point with corresponding119883lowast(119905) = 119883
(119906lowast)(119905) then by the sufficient and necessary
optimality condition (17) in Theorem 4 we get
119864 [119906lowast(119905) | G
119905] = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(44)
Since 119906lowast(119905) isG119905-adapted we have
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
(45)
where 119901(119905) 119902(119905) and 119903(119905 119911) are given by (41)
On the other hand by the sufficient and necessary opti-mality condition (17) in Theorem 4 we obtain
sum
120591119894lt119879
119864 [119878120585lowast
119894+ 120603119901 (120591
119894) minus 120599120582 (120591
119894) | G120591119894] = 0 (46)
That is for each 120591119894lt 119879 we have
119864 [120585lowast
119894| G120591119894] =
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (47)
Since 120585119894is anG
120591119894-measurable random variable we have
120585lowast
119894=
1
119878
119864 [120599120582 (120591119894) minus 120603119901 (120591
119894) | G120591119894] (48)
where 120582(119905) is given by (43) and 119901(119905) is given by (41) Con-sequently we summarize the above results in the followingtheorem
Theorem 8 Let 119901(119905) 119902(119905) and 119903(119905 119911) be the solutions of (41)and let 120582(119905) be the solution of (43) Then the pair (119906lowast(119905) 120585lowast(119905))is given by
119906lowast(119905) = 119864 [120577 (119905) 119901 (119905) + 120587 (119905) 119902 (119905)
+int
R0
984858 (119905 119911) 119903 (119905 119911) ] (119889119911) | G119905]
120585lowast(119905) = sum
119894ge1
120585lowast
1198941[120591119894 119879]
(119905) 119905 le 119879
(49)
where 120585lowast119894given by (48) is the local critical point of the classical
and impulse controls problem (38)
5 Conclusion
We consider the partial information classical and impulsecontrols problem of forward-backward systems driven byLevy processes The control variable consists of two com-ponents the classical stochastic control and the impulsecontrol Because of the non-Markovian nature of the par-tial information dynamic programming principle cannotbe used to solve partial information control problems Asa result we derive a maximum principle for this partialinformation problem Because the concavity conditions of theutility functions and the Hamiltonian process may not holdin many applications we give the sufficient and necessaryoptimality conditions for the local critical points of thecontrol problem To illustrate the theoretical results we usethe maximum principle to solve a portfolio optimizationproblem with piecewise consumption processes and give itsexplicit solutions
In this paper we assume that the two different jumps inour system do not occur at the same time (Assumption 1)This assumption makes the problem easier to analyze How-ever it may fail in many applications Without this assump-tion it requiresmore attention to distinguish between the twodifferent jumpsThiswill be explored in our subsequentwork
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China under Grant 11171050
References
[1] G Mundaca and B Oslashksendal ldquoOptimal stochastic interventioncontrol with application to the exchange raterdquo Journal ofMathematical Economics vol 29 no 2 pp 225ndash243 1998
[2] A Cadenillas and F Zapatero ldquoClassical and impulse stochasticcontrol of the exchange rate using interest rates and reservesrdquoMathematical Finance vol 10 no 2 pp 141ndash156 2000
[3] R R Lumley and M Zervos ldquoA model for investments in thenatural resource industry with switching costsrdquoMathematics ofOperations Research vol 26 no 4 pp 637ndash653 2001
[4] H Meng and T K Siu ldquoOptimal mixed impulse-equity insur-ance control problem with reinsurancerdquo SIAM Journal onControl and Optimization vol 49 no 1 pp 254ndash279 2011
[5] H Meng and T K Siu ldquoOn optimal reinsurance dividend andreinvestment strategiesrdquoEconomicModelling vol 28 no 1-2 pp211ndash218 2011
[6] F Zhang ldquoStochastic differential games involving impulse con-trolsrdquo ESAIM Control Optimisation and Calculus of Variationsvol 17 no 3 pp 749ndash760 2011
[7] L G Wu X J Su and P Shi ldquoOutput feedback controlof Markovian jump repeated scalar nonlinear systemsrdquo IEEETransactions on Automatic Control vol 59 no 1 pp 199ndash2042014
[8] B Oslashksendal and A Sulem Applied Stochastic Control of JumpDiffusions Springer Berlin Germany 2nd edition 2007
[9] R C Seydel ldquoExistence and uniqueness of viscosity solutionsfor QVI associated with impulse control of jump-diffusionsrdquoStochastic Processes and Their Applications vol 119 no 10 pp3719ndash3748 2009
[10] Y Wang A Song C Zheng and E Feng ldquoNonzero-sumstochastic differential game between controller and stopperfor jump diffusionsrdquo Abstract and Applied Analysis vol 2013Article ID 761306 7 pages 2013
[11] Z Wu and F Zhang ldquoStochastic maximum principle for opti-mal control problems of forward-backward systems involvingimpulse controlsrdquo IEEE Transactions on Automatic Control vol56 no 6 pp 1401ndash1406 2011
[12] Z Wu and F Zhang ldquoMaximum principle for stochasticrecursive optimal control problems involving impulse controlsrdquoAbstract and Applied Analysis vol 2012 Article ID 709682 16pages 2012
[13] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010
[14] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-
trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010
[15] X Gao K L Teo andGDuan ldquoAn optimal control approach torobust control of nonlinear spacecraft rendezvous system with120579-119863 techniquerdquo International Journal of Innovative ComputingInformation and Control vol 9 no 5 pp 2099ndash2110 2013
[16] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time TakagindashSugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 21 no 4 pp655ndash671 2013
[17] F Baghery and B Oslashksendal ldquoAmaximumprinciple for stochas-tic control with partial informationrdquo Stochastic Analysis andApplications vol 25 no 3 pp 705ndash717 2007
[18] T T K An and B Oslashksendal ldquoMaximumprinciple for stochasticdifferential games with partial informationrdquo Journal of Opti-mization Theory and Applications vol 139 no 3 pp 463ndash4832008
[19] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward-backward stochastic differential equationswith jumpsrdquo SIAM Journal on Control and Optimization vol48 no 5 pp 2945ndash2976 2009
[20] Z Yu ldquoThe stochastic maximum principle for optimal controlproblems of delay systems involving continuous and impulsecontrolsrdquo Automatica vol 48 no 10 pp 2420ndash2432 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
