Optimal reinsurance under risk and uncertainty

Optimal reinsurance under risk and uncertainty

Alejandro Balb·s,!Beatriz Balb·s,yRaquel Balb·szand Antonio Herasx

Abstract. This paper deals with the optimal reinsurance problem if both insurer and

reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also

included. The insurer and reinsurer degrees of uncertainty do not have to be identical.

The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to

the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the

reinsurer moral hazard is totally eliminated.

Three main contributions seem to be reached. Firstly, necessary and su¢ cient opti-

mality conditions are given. Secondly, the optimal contract is often a bang-bang solution,

i.e., the sensitivity between the retained risk and the total claims saturates the imposedconstraints. For some special cases the optimal contract might not be bang-bang, but there

is always a bang-bang contract as close as desired to the optimal one. Thirdly, the optimal

reinsurance problem is equivalent to other linear programming problem, despite the fact

that risk, uncertainty, and many premium principles are not linear. This may be impor-

tant because linear problems are easy to solve in practice, since there are very e¢ cient

algorithms.

Key Words. Risk and Uncertainty, Moral Hazard, Optimal Reinsurance and Opti-

mality Conditions, Bang-Bang Solution, The Optimal Reinsurance Linear Problem.

A.M.S. Classification Subject. 91B30, 90C48.

JEL Classification. G22.

1. Introduction

Since Borch (1960) and Arrow (1963) published their celebrated seminal papers, theoptimal reinsurance problem has been addressed by many authors and under manydi§erent risk measurement methods and premium principles. Recent approaches are,amongst many others, Kaluszka (2005), Cai and Tan (2007) and Chi and Tan (2013).

!University Carlos III of Madrid. CL. Madrid 126. 28903 Getafe (Madrid, Spain). [email protected]

†University of Castilla la Mancha. Avda. Real F·brica de Seda, s/n. 45600 Talavera (Toledo,Spain). [email protected]

‡University Complutense of Madrid. Department of Actuarial and Financial Economics.Somosaguas-Campus. 28223 Pozuelo de AlarcÛn (Madrid, Spain). [email protected]

§University Complutense of Madrid. Department of Actuarial and Financial Economics.Somosaguas-Campus. 28223 Pozuelo de AlarcÛn (Madrid, Spain). [email protected]

1

Optimal reinsurance under risk and uncertainty 2

Usually, researchers consider the insurer point of view, though the reinsurer viewpointmay be also incorporated (Cai et al., 2012, Cui et al., 2013, etc.). An interestingsurvey about the State of the Art in 2009 may be found in Centeno and Simoes(2009).All the papers above assume that the statistical distribution of claims is known.

Nevertheless, measurement errors or lack of complete information may provoke dis-crepancies between the real and the estimated probabilities of the states of nature,generating uncertain (also called ambiguous) frameworks. Actuarial and Önancial lit-erature is recently paying signiÖcant attention to those cases where the probabilitiesof the scenarios are not totally known. Interesting examples are, among many others,portfolio management (Zhu and M. Fukushima, 2009), equilibrium in asset markets(Bossaerts et al., 2010) and optimal stopping (Riedel, 2009).The Örst objective of this paper is to incorporate ambiguity in the optimal rein-

surance problem, though many results will be also new in the uncertainty free setting.Both insurer and reinsurer may be ambiguous, but their degrees of ambiguity do nothave to be identical. Since the reinsurer information about the reinsured set of policiescould be lower than the information of the insurer, it seems natural to assume thatthe reinsurer ambiguity is higher, but we will not impose this hypothesis becausewe will not need it. According to the empirical evidence and the famous Ellsbergparadox, agents usually reáect ambiguity aversion, so we will accept this assumptionin our analysis. Though there are other recent approaches (Maccheroni et al., 2006),the worst-case principle properly incorporates the ambiguity aversion (Gilboa andSchmeidler, 1989), and therefore our analysis will deal with this principle when con-sidering the insurer expected wealth, the insurer global risk (integrating uncertaintytoo) and the reinsurer premium principle. Actually, all of the papers above deal withambiguity by means of a worst-case approach.Stop-loss or closely related contracts frequently solve the optimal reinsurance

problem. These solutions have been often criticized by both theoretical researchersand practitioners. In practice, reinsurers will rarely accept these solutions due to thelack of incentives of the insurer to verify claims beyond some thresholds. Our secondobjective will be to overcome this caveat. Consequently, the insurer decision variablewill be the (almost everywhere) mathematical derivative of the retained risk with re-spect to the global claims, rather than the retained risk itself. With this modiÖcationwe can impose positive lower bounds to this decision variable, and therefore contractsreáecting spreads with null derivative (áat behavior of the retained risk with respectto the global claims) become unfeasible. In other words, the usual reinsurer moralhazard is eliminated with this approach.The paper is organized as follows. Section 2 will present the general framework,

the set of priors, the properties of the insurer risk measure (integrating uncertainty),the properties of the reinsurance premium principle (which may incorporate the rein-


surer uncertainty) and the general optimal reinsurance problem we are going to dealwith. We will point out how our approach contains most of the usual cases and ex-tends them all if ambiguity arises. Section 3 will be devoted to dealing with two dualapproaches. Theorems 4 and 5 will provide us with two alternative dual problems,as well as two di§erent families of necessary and su¢ cient optimality conditions. Itis worth to point out that one of the duals is linear.The optimality conditions will generate two di§erent ways permitting us to lin-

earize the optimal reinsurance problem. The Örst one is the introduction of a linearoptimization problem generated by a dual solution. This method will allow us toprove Theorem 7 in Section 4, which will show that the optimal contract is in the clo-sure of a set composed of convex combinations of bang-bang contracts, i.e., contractssuch that the derivative of the retained risk with respect to the total one saturatesthe imposed constraints. A clear consequence is that in many classical approachesone must Önd stop-loss or closely related optimal contracts. In our less restrictiveframework the optimal retention will be often a bang-bang solution.Section 5 explores a second linearization procedure. In order to simplify the

exposition the focus is on the Robust Conditional Value at Risk (robust CV aR) as aninsurer risk/ambiguity measure and a reinsurer instrument to generate the premiumprinciple. The method applies for much more situations, but selecting one importantcase we signiÖcantly shorten the paper. Furthermore, the CV aR is becoming moreand more popular among researchers and practitioners due to its interesting properties(Ogryczak and Ruszczynski, 2002).Since one of the two duals of Section 3 is linear, we will construct the double-

dual (dual of the dual) optimal reinsurance problem in Section 5, which is linear.We will prove that the solution of the double-dual will directly lead to the optimalreinsurance contract. This seems to be a very important property because thereare many e¢ cient algorithms solving linear problems in both Önite-dimensional andinÖnite-dimensional frameworks (Anderson and Nash, 1987). Besides, linear problemsoften lead to extreme solutions, which explains why the non linear optimal reinsuranceproblem may be solved by a bang-bang retention.The last section of the paper summarizes the most important conclusions, em-

phasizing the two main novelties (uncertainty introduction and moral hazard elimina-tion) and the three main contributions (necessary and su¢ cient optimality conditions,bang-bang solutions and double-dual linear problems).Throughout the paper we will need several mathematical notions about topolog-

ical spaces, Banach and Hilbert spaces, strong and weak convergences, etc. Someof them will be brieáy summarized, but further discussions may be found in Luen-berger (1969), Kelly (1975), Rudin (1973) and (1987), or Anderson and Nash (1987),amongst others.

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��IP� (yz) � (y)

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5>A 4E4AH y � L2 (��0)� �DAC74A<>A4� �� 7>;3B 85 0=3 >=;H 85 8B 2>=C8=D>DB�BD1�0338C8E4 � (y1 + y2) � (y1) + (y2) 5>A yi � L

2 (��0)� i = 1, 2� 0=3 7><>64�=4>DB � (�y) = � (y) 5>A � � 0 0=3 y � L2 (��0)�� *4 F8;; 0;B> 8<?>B4 C> 14E��CA0=B;0C8>= 8=E0A80=C 5>A B><4 E� � �� (y + k) = (y) � E�k 85 k � �� 0=3y � L2 (��0)�� C 8B 4@D8E0;4=C C> C74 5D;J;;<4=C >5

��IP� (z) = E� ��

5>A 4E4AH z � �� &D<<0A8I8=6� F4 70E4�

��* �%�! �� -< E� � �� *- ) :-)4 6=5*-:� : L2 (��0) �� 1; ;)1, <7 *- )E��<:)6;4)<176 16>):1)6< :1;3 5-);=:- 1. <0- <?7 -9=1>)4-6< +76,1<176; *-47? 074,�a) : L2 (��0) �� 1; 67:5�+76<16=7=;� E��<:)6;4)<176 16>):1)6<� ;=*�),,1<1>-

)6, 0757/-6-7=;�b) &0-:- -@1;<; ) +76>-@ )6, ?-)34A�+758)+< ;-< �� L2 (��0) ;=+0 <0)< ��

074,; .7: ->-:A y � L2 (��0) )6, �� 074,; .7: ->-:A z � ��. �

*4 F8;; =>C ?A>E4 C74 4@D8E0;4=24 14CF44= �>=38C8>=B a) 0=3 b) 01>E4 1420DB40=0;>6>DB ?A>>5B <0H 14 5>D=3 8= B4E4A0; ?0?4AB �5>A 8=BC0=24� B44 �0;1RB et al.,�� ">C824 C70C �� 8<?;84B C70C 8B F40:;H ;>F4A B4<8�2>=C8=D>DB�

'74A4 0A4 <0=H A8B: <40BDA4B 8= ?A02C824� �2CD0;;H� 4E4AH 4G?42C0C8>= 1>D=343A8B: <40BDA4 0=3 4E4AH 34E80C8>= <40BDA4 �%>2:054;;0A et al�� B0C8B5H �4J=8C8>=1 01>E4 F8C7 E� = 1 0=3 E� = 0� A4B?42C8E4;H� '74 2>74A4=C A8B: <40BDA4B >5 �ACI=4Aet al� �� 0;B> B0C8B5H �4J=8C8>= 1 F8C7 E� = 1� �B B083 8= C74 8=CA>3D2C8>=�C74 �;;B14A6 ?0A03>G 8=3820C4B C74 ?A4B4=24 >5 0<186D8CH 0E4AB8>=� F7827 <0H 148=2>A?>A0C43 1H 340;8=6 F8C7 C74 F>ABC�20B4 ?A8=28?;4 ��8;1>0 0=3 &27<483;4A� �� 2>=B8BC4=C F0H C> 2>=B834A C78B ?A8=28?;4 8B C> 34J=4 MC74 A>1DBC 4GC4=B8>= >5 0A8B: <40BDA4 A4;0C8E4 C> C74 B4C >5 ?A8>AB P0UN� �>A 8=BC0=24� C74 A>1DBC CV aR F8C72>=J34=24 ;4E4; 0 � � < 1 A4;0C8E4 C> C74 B4C >5 ?A8>AB P0U F8;; 14 68E4= 1H

RCV aR(�U ,�)(y) :=Max

�CV aR(p,�) (y) ; p � P

0U

��

5>A 4E4AH y � L2 (��0)� CV aR(p,�) (y) 34=>C8=6 C74 DBD0; CV aR >5 y 85 p 8B C74 B4;42C43?A>1018;8CH <40BDA4 0=3 � 8B C74 ;4E4; >5 2>=J34=24� '74>A4< 2 14;>F B7>FB C70CC78B 34J=8C8>= 8B 2>=B8BC4=C�

��!#�� %=887;- <0)< 0 � � < 1� &0-6� CV aR(p,�) (y) -@1;<; .7: ->-:A p � P0U

)6, ->-:A y � L2 (��0)� <0- 5)@15=5 16 �� -@1;<; .7: ->-:A y � L2 (��0)� )6,RCV aR(�U ,�)

;)<1;C-; �-C61<176 1b) ?1<0 ERCV aR(P�U ,�)

= 1 )6,

�RCV aR(P�U ,�)

=

�z � L2 (��0) ; ��IP� (z) = 1 and �f � PU with 0 � z �

f1��

�.. ��

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

�6 8):<1+=4):� �� 4-),; <7��

��

RCV aR(�U ,�)(y) =Max � ��IP� (yz)

��IP� (f) = 1

��IP� (z) = 1

0 � z � f1��, f � R

z � L2 (��0) and f � L2 (��0) are decision variables

��

.7: ->-:A y � L2 (��0)�

�#!!�� 8ABC ;4C 0B ?A>E4 C70C C74 B4C�RCV aR(P�U ,�)

68E4= 1H �� 8B 2>=E4G� �=3443�

BD??>B4 C70C z1 0=3 z2 14;>=6 C> C78B B4C 0=3 C0:4 0 � � � 1� �5 f1 0=3 f2 0A4 C74>1E8>DB <40BDA01;4 5D=2C8>=B 68E4= 1H �� C74= z = �z1 + (1� �) z2 0=3 f =�f1 + (1� �) f2 CA8E80;;H B0C8B5H C74 2>=38C8>=B 6D0A0=C448=6 C70C z � �RCV aR

(P�U ,�)�


8B =>A<�1>D=343� �=3443� �� 8<?;84B C70C �z�(2,IP�) �� R1��

��(2,IP�)

5>A 4E4AH z � �RCV aR(P�U ,�)

�


2;>B43� �=3443� BD??>B4 C70C C74 B4@D4=24 (zn)�

n=1 � �RCV aR(P�U ,�)

2>=E4A64B C> z 8= L2 (��0) 0=3 ;4C DB ?A>E4 C70C z � �RCV aR(P�U ,�)

� 4C (fn)�

n=1 �

L2 (��0) C74 >1E8>DB B4@D4=24� &8=24 (fn)�

n=1 � PU 0=3 C78B B4C 8B F40:;H 2><?02C�C74A4 4G8BCB f � PU F7827 8B 0 F40: 066;><4A0C8>= ?>8=C� '7DB (z, f) 8B 0 F40:066;><4A0C8>= ?>8=C >5 (zn, fn)

�

n=1 0=3 C74A45>A4 0 � z �f1��, f � R CA8E80;;H 5>;;>F

5A>< 0 � zn �fn1��, fn � R 5>A 4E4AH n � ��


8B F40:;H 2><?02C� �=3443��RCV aR(P�U ,�)

8B F40:;H 2;>B43 1420DB4 8C

8B 2;>B43 0=3 0=3 2>=E4G ��07=��0=027 '74>A4<�� B> C74 �;0>6;DLB '74>A4< 8<?;84BC70C �RCV aR

(P�U ,�)8B F40:;H 2><?02C 1420DB4 8C 8B =>A<�1>D=343�


148=6 2>=E4G 0=3 F40:;H 2><?02C� F4 70E4 C70C C74 <0G8<D< 8=

C74 A867C 70=3 B834 >5 �� >1E8>DB;H 4G8BCB� �4=>C4 1H (y) C78B <0G8<D<� �C 8B:=>F= �%>2:054;;0A et al�� C70C 5>A p � P0U

CV aR(p,�) (y) =Max

��p (yz) ; ��p (z) = 1, 0 � z �

1

1� �

��

7>;3B F74=4E4A y 8B 8=C46A01;4 F8C7 A4B?42C C> p� �5 y � L2 (��0) C74= |y|R 8B 8=C46A01;4F8C7 A4B?42C C> ��0 0=3

��p (|y|) =

� M

0

|y| dp =

� M

0

|y|dp

d��0d��0 �

� M

0

|y|Rd��0 <

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1

5>A 4E4AH p � P0U � B> y 8B 8=C46A01;4 F8C7 A4B?42C C> p 0=3 CV aR(p,�) (y) 4G8BCB�'74 C74>A4< F8;; 14 >1E8>DB;H ?A>E43 85 F4 B7>F C70C (y) 4@D0;B C74 A867C 70=3

B834 >5 � �� 4=>C4 1H RCV aR(�U ,�)(y) � C74 BD?A4<D< 8= C74 A867C 70=3 B834

>5 � �� 5 p � P0U �� 8<?;84B C74 4G8BC4=24 >5 0 � z � 11��

F8C7

1 = ��p (z) = ��IP�

zdp

d��0

0=3

CV aR(p,�) (y) = ��p (yz) = ��IP�

yzdp

d��0

.

z = z dpdIP�

� �RCV aR(P�U ,�)

1420DB4 f = dpdIP�

B0C8BJ4B C74 A4@D8A43 2>=38C8>=B� �4=24�

CV aR(p,�) (y) = ��IP� (yz) � (y) 0=3 C74A45>A4 RCV aR(�U ,�)(y) � (y)�

'74 >??>B8C4 8=4@D0;8CH 0;B> 7>;3B� �=3443� 85 (y) = ��IP� (yz) F8C7 z ��RCV aR

(P�U ,�)� C74= C0:4 C74 >1E8>DB f 0=3 C74 ?A>1018;8CH <40BDA4 dp = fd��0 B0C�

8BJ4B p � P0U � !>A4>E4A� z =zf�C0:4 0

0= 0� B0C8BJ4B 0 � z � 1

1��0=3

��p (z) =

� M

0

zdp =

� M

0

zfd��0 =

� M

0

zd��0 = ��IP� (z) = 1,

B>RCV aR(�U ,�)

(y) � CV aR(p,�) (y) � ��p (yz) = ��M0yzdp =

��M0yzfd��0 = �

�M0yzd��0 = ��IP� (yz) = (y) .

�

��#� �� &0- +76;<)6< :)6,75 >):1)*4- z = 1 *-476/; <7 �RCV aR(P�U ,�)

*-+)=;-

f = 1 ;)<1;C-; <0- +76,1<176; 7. � �� -6+-� �� ;07?; <0)< RCV aR(�U ,�)(y) �

��P� (y) .7: ->-:A y � L2 (��0)� i.e.� RCV aR(�U ,�)

1; -@8-+<)<176 *7=6,-, 16 <0-

;-6;- 7. $7+3).-44): et al� ��

��#� �� z � �CV aR�IP�,��=� z � �RCV aR

(P�U ,�)*-+)=;- f = 1 ;)<1;C-; <0-

:-9=1:-, +76,1<176;� �7:-7>-:� �. R � 1 1; +76;<)6< <0-6

0 � z �f

1� �, f � R =� 0 � z �

R

1� �=

1

1� �

?1<0 � = 1��1��R

�� &0-:-.7:-� �� )6, �� <:1>1)44A 4-), <7

CV aR(IP�,�) (y) � RCV aR(�U ,�)(y) � CV aR(IP�,�) (y) .

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1

�< 1; -);A <7 ;-- <0)< 0 � 1 � 1R� � < 1 )6, � � �� 6 7<0-: ?7:,;� 1. R � 1 1;

+76;<)6< <0-6 <0- :7*=;< CV aR 4)A; ?1<016 <0- ;8:-), 7. <?7 CV aRs� �

��#� �� %1514):4A� 76- +)6 ,-C6- ) :7*=;< -@<-6;176 :-4)<1>- P0U 7. 5)6A +7�0-:-6< 7: -@8-+<)<176 *7=6,-, :1;3 5-);=:-;� ;=+0 ); <0- ()6/ 5-);=:-� <0- 87?-:<:)6;.7:5� <0- ?-1/0<-, CV aR� -<+� �44 7. <0-5 ?144 ;)<1;.A �-C61<176 1 ?1<0 E� = 1�$7*=;< ,->1)<176; ;)<1;.A16/ E� = 0 5)A *- )4;7 ,-C6-,� �7: 16;<)6+-� <0- :7*=;<)*;74=<- ,->1)<176 ?144 *- /1>-6 *A

RAD�U(y) :=Max

��p (|y � ��p (y)|) ; p � P

0U

�,

)6, <0- :7*=;< ;<)6,):, ,->1)<176 ?144 *-

RSD�U(y) :=Max

��p

�(y � ��p (y))

2�; p � P0U�.

(- ?144 67< )6)4AB- <0- ,->1)<176; )*7>- *-+)=;- <0-A +)6 *- ;<=,1-, ?1<0 <0- 5-<0�7,; 7. &0-7:-5 2� �

�� "#��&� "#� ��"�� 5 x � L2 [0,M ] 8B C74 A4C08=43 A8B: 0=3 1� x �L2 [0,M ] 8B C74 24343 >=4� C74= A48=BDA0=24 ?A824 >5 F8;; 14 68E4= 1H

Re insurance/Pr ice = � (J (1� x)) , ��

� : L2 (��0) �� 148=6 0 E��CA0=B;0C8>= 8=E0A80=C A8B: <40BDA4 5>A B><4 E� � ��4J=8C8>= 1�� ">C824 C70C �� 2>=C08=B C74 DBD0; ?A4<8D< ?A8=28?;4B� �=3443� 5>A 0;8=40A ?A8=28?;4

� (J (1� x)) = ��IP� (J (1� x) c) , ��

F8C7 c � L2 (��0) 0=3 c � 0� 8C 8B >1E8>DB C70C � B0C8BJ4B �4J=8C8>= 1b) F8C7 �� ={�c} 0=3 E� = ��IP� (c)� �= ?0AC82D;0A� C74 �G?42C43 )0;D4 $A4<8D< $A8=28?;4�EV PP �

Re insurance/Pr ice = (1 + �) ��IP� (J (1� x)) ,

� � 0 148=6 C74 ;>038=6 A0C4� 8B 0 ;8=40A ?A8=28?;4 F8C7 c = 1 + �� 0=3 C74A45>A4 8C8B 8=2;D343 8= �� B B083 01>E4� 4E4AH 34E80C8>= <40BDA4 0=3 4E4AH 4G?42C0C8>=1>D=343 >A 2>74A4=C A8B: <40BDA4 B0C8BJ4B �4J=8C8>= 1� 0=3 C74A45>A4 B> 3>4B

� (J (1� x)) = ��IP� (J (1� x)) + � (�J (1� x)) , ��

�, � � 0� F8C7�� = {�� z; z � ��} ��

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

0=3 E� = �� E� �B44 �� >=B4@D4=C;H� <0=H 2;0BB820; =>=�;8=40A ?A4<8D<?A8=28?;4B 0A4 8=2;D343 8= �� *0=6 $A8=28?;4� &C0=30A3 �4E80C8>= $A4<8D< $A8=�28?;4� 4C2�� !>A4 8<?>AC0=C;H� �� 0;B> 8=2;D34B ?A4<8D< ?A8=28?;4B 8=E>;E8=6 C74A48=BDA4A 0<186D8CH B8=24� 5>A 8=BC0=24� A>1DBC 4GC4=B8>=B >5 C74 CV aR� C74 F4867C43CV aR� C74 *0=6 <40BDA4 >A C74 BC0=30A3 34E80C8>= <0H ?;0H C74 A>;4 >5 8= ��!>A4>E4A� � <0H 14 A4;0C43 C> C74 0<186D8CH ;4E4; >5 C74 A48=BDA4A� F7827 3>4B =>C=424BB0AH 4@D0; C74 8=BDA4A 0<186D8CH ;4E4;� �>A 8=BC0=24� 8C <0H <0:4 B4=B4 C> 0B�BD<4 C70C C74 A48=BDA4A 0<186D8CH ;4E4; 8B 78674A� 0=3 C74A45>A4 8C 8B 270A02C4A8I43 1H

C74 B4C >5 ?A8>AB�f � L2 (��0) ; 0 � f � R, ��IP� (f) = 1

�F8C7 R � R �B44 � ��

�� !�� "#!�� '74 ?A4B4=24 >5 0<186D8CH 0E4AB8>= 0=3 C74 F>ABC�20B4 ?A8=28?;4 ��8;1>0 0=3 &27<483;4A� �� <0H 0;B> 9DBC85H C74 8=CA>3D2C8>= >5 MC74A>1DBC 4G?42C0C8>= A4;0C8E4 C> C74 B4C >5 ?A8>AB PUN14;>F

��U (W ) :=Min��p (W ) ; p � P

0U

�=Min {��IP� (Wf) ; f � PU} ��

F74A4 W � L2 (��0) 8B C74 �D=24AC08=� 8=BDA4A F40;C7 0C T � PU 8B C74 B4C >5 540B81;4?A8>AB 0=3 ��p (W ) = ��IP� (Wf) 8B C74 4G?42C0C8>= >5 W D=34A dp = fd��0� ">C824C70C C74 <8=8<D< 8= �� 4G8BCB 1420DB4

L2 (��0) � f � ��IP� (Wf) � ��

8B 0 F40:;H 2>=C8=D>DB 5D=2C8>= 5>A 4E4AH W � L2 (��0) 0=3 PU 8B 0 F40:;H 2><?02CB4C �*484ABCA0BB '74>A4<�� #1E8>DB;H� �� ;403B C>

��U (W ) =Max {��p (W ) ; p � P} =Max {��IP� (Wf) ; f � PU} , ��

5>A 4E4AHW � L2 (��0)� �= >C74A F>A3B� ��U 0;B> B0C8BJ4B �4J=8C8>= 1 F8C7 E�IEPU =1 0=3 ��IEPU

= PU �&D??>B4 C70C A 34=>C4B C74 8=2><4 64=4A0C43 1H C74 B>;3 8=BDA0=24 ?>;8284B�

�40A8=6 8= <8=3 C74 A48=BDA0=24 2>=CA02C 0=3 �� C74 8=BDA4A F40;C7 0C T F8;; 14

A� J (x)�� (J (1� x)) . ��

'74 8=BDA4A <0H B4;42C 0 A8B: <40BDA4 ��4J=8C8>= 1� 8= >A34A C> 2>=CA>; C74A8B: >5 �� '74 >?C8<0; A48=BDA0=24 ?A>1;4< F8;; <0G8<8I4 C74 A>1DBC 4G?42C43F40;C7 ��U �F7827 8B 4@D8E0;4=C C> C74 <8=8<8I0C8>= >5 ��U � 0=3 F8;; <8=8<8I4 �'7DB� 140A8=6 8= <8=3 C74 ?A>?4AC84B >5 ��U 0=3 ��4J=8C8>= 1�� C74 >?C8<8I0C8>=?A>1;4< 142><4B��

�

Min (A� J (x)�� (J (1� x))) = (�J (x)) + E�� (J (1� x))� E�AMin � ��U (A� J (x)�� (J (1� x))) = ��U (�J (x)) +� (J (1� x))� AH � x � 1

,

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

x � L2 [0,M ] 148=6 C74 3428B8>= E0A801;4� &8=24 E� 0=3 A 0A4 2>=BC0=C F4 20= A4<>E4E�A 0=3 A 8= C74 E42C>A >?C8<8I0C8>= ?A>1;4< 01>E4 F8C7>DC 0;C4A8=6 C74 $0A4C>B>;DC8>=B� !>A4>E4A� B8=24 �� 0=3 �� B7>F C70C �� 0=3 0A4 2>=E4G 5D=2C8>=B>= L2 (��0)� 4E4AH ?A>?4A $0A4C> >?C8<D< <0H 14 >1C08=43 1H <8=8<8I8=6 0 B20;0A>1942C8E4

(�J (x)) + E�� (J (1� x)) + w (��U (�J (x)) +� (J (1� x))) ,

w > 0 148=6 0 MF4867CN5>A ��U �"0:0H0<0 et al.� �� 2CD0;;H� w 8=3820C4B C74A4;0C8E4 8<?>AC0=24 >5 C74 A>1DBC 4G?42C43 A4CDA= F8C7 A4B?42C C> C74 A8B: �C74 78674AC74 E0;D4 >5 w C74 78674A C74 8<?>AC0=24 >5 ��U 5>A C74 8=BDA4A�� !0=8?D;0C8=6� C74>1942C8E4 5D=2C8>= 142><4B

(�J (x)) + w (�� (�J (x))) +�E� + w

�� (J (1� x))

0=3 C74 >?C8<0; A48=BDA0=24 ?A>1;4< F8;; 14 Min (�J (x)) + w (��U (�J (x))) +

�E� + w

�� (J (1� x))

H � x � 1, ��

x � L2 [0,M ] 148=6 C74 3428B8>= E0A801;4�

3. D/�& �**,)��"

4C DB 8=CA>3D24 0= 8=BCAD<4=C0; M!40= )0;D4 '74>A4<N F7>B4 ?A>>5 8B ><8CC431420DB4 0 B8<8;0A >=4 <0H 14 5>D=3 8= �0;1RB et al� ��

�� %=887;- <0)< Y 1; ) �)6)+0 ;8)+-� Z 1; 1<; ,=)4� )6, K � Z 1; ) +76>-@)6, (Z, Y )�+758)+< ;-<� %=887;- <0)< � 1; )6 166-: :-/=4): 8:7*)*141<A 5-);=:- 76<0- �7:-4 �)4/-*:) 7. K -6,7?-, ?1<0 <0- (Z, Y )�<78747/A� &0-6� <0-:- -@1;<;) =619=- k� � K ;=+0 <0)<

�

K

y, z d� (z) = y, k�

074,; .7: ->-:A y � Y. �

4C DB 8=CA>3D24 CF> 3D0; ?A>1;4A �� 0B F4;; 0B =424BB0AH 0=3 BDP 284=C�0ADB7��D7=�'D2:4A ;8:4 >?C8<0;8CH 2>=38C8>=B� &C0=30A3 3D0;8CH C74>AH 3>4B =>C6D0A0=C44 C74 01B4=24 >5 3D0;8CH 60? 85 8=J=8C4�38<4=B8>=0; ?A>1;4;E43��D0;8CH 60?B <0H >22DA 4E4= 8= ;8=40A ?A>6A0<<8=6 ��=34AB>= 0=3 "0B7� �� =>A34A C> >E4A2><4 C78B 20E40C� JABC >5 0;; F4 F8;; 68E4 0 2>=20E4 3D0; B0C8B5H8=6 C74

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1

&;0C4A @D0;8J20C8>= � D4=14A64A� �� F7827 6D0A0=C44B C74 01B4=24 >5 3D0;8CH 60?�'74=� F4 F8;; 68E4 0 ;8=40A 3D0; =>C B0C8B5H8=6 C74 &;0C4A @D0;8J20C8>=� "4E4AC74;4BB�C74 01B4=24 >5 3D0;8CH 60? F8C7 C74 2>=20E4 3D0; F8;; 0;;>F DB C> ?A>E4 C74 01B4=24>5 3D0;8CH 60? F8C7 C74 ;8=40A >=4 C>>�

��!#�� 76;1,-: ":7*4-5��

��

Max

��

��

��E� + w

� 1, J�(�)

+ H, J��z + wf +

�E� + w

��+

� 1, J��z + wf +

�E� + w

��

f � PU , z � ��, � � ��

, ��

(f �, z�, ��) � PU ×�� ×�� *-16/ <0- ,-+1;176 >):1)*4-� &0-6�

a) &0- 51615=5 7. � � )6, <0- 5)@15=5 7. �� ):- C61<-� )<<)16)*4- )6,1,-6<1+)4�b) %=887;- <0)< x� 1; � ��.-);1*4- )6, (f �, z�, ��) 1; ��.-);1*4-� &0-6� x� ;74>-;

� � )6, (f �, z�, ��) ;74>-; �� 1. )6, 764A 1.� 7=< 7. �-*-;/=- 6=44 ;-<; 7. [0,M ]�

��

��

��IP� (J (x�) f �) � ��IP� (J (x

�) f) , �f � PU��IP� (J (x

�) z�) � ��IP� (J (x�) z) , �z � ��

��IP� (J (1� x�) ��) � ��IP� (J (1� x

�) �) , ��

x� (t) = H (t) , J��z� + wf � +

�E� + w

��(t) > 0

x� (t) = 1, J��z� + wf � +

�E� + w

��(t) < 0

��

�#!!�� 4C DB ?A>E4 C70C �� 8B B>;E01;4� �=3443 C74 540B81;4 B4C H � x � 1 8BF40:;H�2><?02C 1420DB4 L2 [0,M ] 8B 0 �8;14AC B?024 0=3 C74A45>A4 A4K4G8E4 ��;0>6;DLB'74>A4<�� 0=3 C74 >1942C8E4 5D=2C8>= >5 �� 8B F40:;H ;>F4A B4<8�2>=C8=D>DB 1420DB48C 8B 0 2><?>B8C8>= >5 C74 (L2 [0,M ] , L2 [0,M ]) � (L2 (��0) , L

2 (��0))�2>=C8=D>DB5D=2C8>= J 0=3 C74 (L2 (��0) , L

2 (��0))�;>F4A B4<8�2>=C8=D>DB 5D=2C8>=B � �� 0=3�� '74A45>A4� �� 0CC08=B 0= >?C8<0; E0;D4 �*484ABCA0BBL'74>A4<��

�B DBD0;� �+ (t) = Maxt {� (t) , 0} 0=3 �� (t) = Maxt {�� (t) , 0} 5>A E4AH � � L

2 [0,M ] 0=34E4AH 0 � t �M � �C 8B 40BH C> B44 C70C � = �+�� 0=3 �+,�� L2 [0,M ]� &8<8;0A =>C0C8>=B <0H14 DB43 8= B8<8;0A B8CD0C8>=B�

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

�G?A4BB8>=B �� F7827 0;B> 0??;84B 5>A � D=34A C74 >1E8>DB <>38J20C8>=B� 0=3�� B7>F C70C $A>1;4< �� 8B 4@D8E0;4=C C> $A>1;4<

��

��

Min �1 + w�2 +�E� + w

��3

�1 � ��IP� (J (x) z) , �z � ��

�2 � ��IP� (J (x) f) , �f � PU�3 � ��IP� (J (x� 1) �) , ��

H � x � 1, �1, �2, �3 � ��

, ��

(�1, �1, �3, x) � �� × �� × �� × L2 [0,M ] 148=6 C74 3428B8>= E0A801;4� '74 JABC� B42>=3

0=3 C78A3 2>=BCA08=CB >5 �� 0A4 E0;D43 8= C74 B?024B >5 2>=C8=D>DB 5D=2C8>=B >= C74F40:;H 2><?02C B?024B K = �� K = PU 0=3 K = �� A4B?42C8E4;H� �>;;>F8=6 D4=14A64A �� C74 0BB>280C43 3D0; E0A801;4B (�1, �2, �3) <DBC 14 8==4A A46D;0A=>=�=460C8E4 <40BDA4B >= C74 �>A4; �0;641A0B >5 C74 2><?02C B4CB 01>E4� 0=3 C74 06A0=680= 5D=2C8>= 8B

L (�1, �1, �3, x, �1, �2, �3) = �1

�1�

��d�1

�+

w�2

�1�

�Ud�2

�+�E� + w

��3

�1�

��d�3

�+

��IP� (J (x) z) d�1 (z) + w

�U��IP� (J (x) f) d�2 (f)+

�E� + w

� ��IP� (J (x) �) d�3 (�)�

�E� + w

� ��IP� (J (1) �) d�3 (�) .

��

&8=24 (�1, �2, �3) 8B 3D0;�540B81;4 85 0=3 >=;H 85

Inf {L (�1, �1, �3, x, �1, �2, �3) ; H � x � 1, �1, �1�3,� ��} > �

� D4=14A64A� �� 1� �2 0=3 �3 <DBC 14 ?A>1018;8CH <40BDA4B �C>C0; <0BB 4@D0; C>>=4�� 0=3 C74 3D0; >1942C8E4 142><4B �B44 ��

��

��

Inf

��

��

��IP� (J (x) z) d�1 (z) + w

�U��IP� (J (x) f) d�2 (f)+

�E� + w

� ��IP� (J (x) �) d�3 (�)

��E� + w

� ��IP� (J (1) �) d�3 (�)

H � x � 1

.

�B 8= �0;1RB et al� �� 4<<0 3 F8C7 Y = Z=L2 (��0) 6D0A0=C44B C70C C74 3D0;B>;DC8>= 8B 02784E43 8= 0 E42C>A >5 C7A44 �8A02 �4;C0B� B> C74 3D0; E0A801;4 <0H B8<?;85H

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

C> (z, f, �) � �� ×PU ×�� 0=3 C74 3D0; >1942C8E4 142><4B �B44 �� 0=3 ��

� (z, f, �) = ��E� + w

�+ ��IP� (J (1) �)+

InfH�x�1

��IP� (J (x) z) + w��IP� (J (x) f) +

�E� + w

��IP� (J (x) �)

�

= ��E� + w

��IP� (J (1) �)+

InfH�x�1

� x, J� (z) +w x, J� (f) +

�E� + w

� x, J� (�)

�

= ��E� + w

� 1, J� (�) +InfH�x�1

�x, J�

�z + wf +

�E� + w

��.

��'74 3D0; ?A>1;4< 8B �

Max � (z, f, �)(z, f, �) � �� × PU ×��

��

'74 01B4=24 >5 3D0;8CH 60? 14CF44= �� >A �� 0=3 �� 0=3 C74 B>;E018;8CH >5�� 7>;3 3D4 C> C74 &;0C4A 2>=38C8>= � D4=14A64A� �� F7827 8B CA8E80;;H 5D;J;;431H ��

'74 8=J<D< >5 �� 8B >1E8>DB;H 02784E43 0C

x (t) =

��

��

H (t) , if J��z + wf +

�E� + w

��(t) > 0

H (t) � x (t) � 1, if J��z + wf +

�E� + w

��(t) = 0

1, if J��z + wf +

�E� + w

��(t) < 0

��

0=3 4@D0;B

�

J�(z+wf+(E�+w)�)(t)>0

�HJ�

�z + wf +

�E� + w

��dt+

�

J�(m+wp+(E�+w)�)(t)<0

J��z + wf +

�E� + w

��dt+

= H,Maxt

�J��z + wf +

�E� + w

��(t) , 0

�

� 1,Maxt

��J�

�z + wf +

�E� + w

��(t) , 0

� .

'7DB� �� 0=3 �� ;403 C> C74 3D0; >1942C8E4 68E4= 8= �� 0=3 a) 8B ?A>E43�

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

�= >A34A C> ?A>E4 b)� >F8=6 C> �� C74 =424BB0AH 0=3 BDP 284=C ?A8<0; 0=3 3D0;>?C8<0;8CH 2>=38C8>=B 5>A �� 142><4 � D4=14A64A� ��

��

��

��1 = ��IP� (J (x�) z�)

��1 � ��IP� (J (x�) z) , �z � ��

��2 = ��IP� (J (x�) f �)

��2 � ��IP� (J (x�) f) , �f � PU

��3 = ��IP� (J (x� � 1) ��)

��3 � ��IP� (J (x� � 1) �) , ��

x� (t) = H (t) , if J��z� + wf � +

�E� + w

��(t) > 0

x� (t) = 1, if J��z� + wf � +

�E� + w

��(t) < 0

,

0=3 �� 142><4B >1E8>DB� �

��#� �� -;81<- <0- .)+< <0)< � � 1; +76>-@ *=< 67< 416-):� ?- ?144 ;-- <0)< 1< 1;+47;-4A :-4)<-, <7 ;->-:)4 416-): 8:7*4-5;� �1:;<4A� 4-< =; ;-- <0)< �� 5)A *- -);14A<:)6;.7:5-, 16<7 ) 416-): 8:7*4-5� )< 4-);< .7: )88:78:1)<- +071+-; 7. PU � )6, �� <1; 1587:<)6< .7: <?7 :-);76;� �1:;<4A� <0-:- ):- 5)6A )4/7:1<05; <7 ;74>- 16 8:)+<1+-*7<0 C61<-�,15-6;176)4 )6, 16C61<-�,15-6;176)4 416-): 8:7*4-5; ��6,-:;76 )6, );0�� %-+76,4A� 416-): 8:7*4-5; 7.<-6 8:-;-6< -@<:-5- �7: *)6/�*)6/� ;74=<176; x�

;=+0 <0)< x� (t) = H (t) 7: x� (t) = 1 5=+0 074, 7=< 7. �-*-;/=- 6=44 ;-<;� &01; +7=4,-@84)16 ?0A <0- ;<78 47;; :-16;=:)6+-� ;)<1;.A16/ x� (t) = 0 7: x� (t) = 1� .:-9=-6<4A;74>-; � � 1. H = 0� �

��!#�� 76;1,-: ":7*4-5��

��

Max H,�d � 1,�u ��E� + w

� 1, J�(�)

J��z + wf +

�E� + w

�� d + �u = 0

�d � 0, �u � 0f � PU , z � ��, � � ��, �d � L

2 [0,M ] , �u � L2 [0,M ]

��

(f, z, �,�d,�u) � PU ×�� ×�� × L2 [0,M ]× L2 [0,M ] *-16/ <0- ,-+1;176 >):1)*4-�

a) &0- 51615=5 7. � � )6, <0- 5)@15=5 7. �� ):- C61<-� )<<)16)*4- )6,1,-6<1+)4�b) %=887;- <0)< x� 1; � ��.-);1*4- )6, (f �, z�, ��,��d,�

�u) 1; ��.-);1*4-� &0-6�

<0-A ;74>- � � )6, �� :-;8-+<1>-4A� 1. )6, 764A 1.��

��

��IP� (J (x�) f �) � ��IP� (J (x

�) f) , �f � PU��IP� (J (x

�) z�) � ��IP� (J (x�) z) , �z � ��

��IP� (J (1� x�) ��) � ��IP� (J (1� x

�) �) , ��

��d (x� �H) = 0

��u (1� x�) = 0

��

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

�#!!�� B 8= C74 ?A>>5 >5 '74>A4< 4� �� 8B 4@D8E0;4=C C> �� '74 06A0=680= >5�� <0H 0;B> 8=2>A?>A0C4 C74 <D;C8?;84AB �d� �u � L

2 [0,M ]� �d � 0� �u � 0� A4;0C43C> C74 2>=BCA08=CB H � x � 1� �= BD27 0 20B4 C74 06A0=680= 142><4B

L� (�1, �1, �3, x, �1, �2, �3,�d,�u) =L (�1, �1, �3, x, �1, �2, �3)+ H � x,�d + x� 1,�u ,

F74A4 L 8B 68E4= 1H �� 4B834B� C74 3D0; E0A801;4 <0H 14 B8<?;8J43 0608= 3D4 C> 4<<0 3 �B44 �� 0=3 C74 06A0=680= >5 �� 142><4B �B44 �� 0=3 ��

L� (x, , z, f, �,�d,�u) = ��E� + w

��IP� (J (1) �) + ��IP� (J (x) z)+

w��IP� (J (x) , f) +�E� + w

��IP� (J (x) , �)+ H � x,�d + x� 1,�u

= ��E� + w

� 1, J� (�) � 1,�u +

x, J��z + wf +

�E� + w

�� d + �u

� + H,�d .

'74 8=J<D< >5 L� (x, , z, f, �,�d,�u) 5>A x � L2 [0,M ] 8B >1E8>DB;H J=8C4 85 0=3 >=;H

85 C74 JABC 2>=BCA08=C >5 �� 7>;3B� 8= F7827 20B4 8C 4@D0;B

��E� + w

� 1, J� (�) � 1,�u + H,�d .

'7DB� �� 8B 0 3D0; ?A>1;4< >5 �� 0=3 �� D4=14A64A� �� '74 &;0C4A @D0;8�J20C8>= 3>4B =>C 7>;3 1420DB4 2>=BCA08=CB H � x 0=3 x � 1 0A4 E0;D43 8= L2 [0,M ]�0=3 C74 ?>B8C8E4 2>=4 >5 C78B B?024 70B E>83 8=C4A8>A� '7DB� 0 3D0;8CH 60? 14CF44=�� 0=3 �� <867C 4G8BC� 0=3 C74 BD?A4<D< >5 �� <867C 14 BCA82C;H ;>F4A C70= C74��>?C8<0; E0;D4� "4E4AC74;4BB� 85 F4 ?A>E4 C70C C74 >?C8<0; E0;D4B >5 �� 0=3 ��0A4 A402701;4 0=3 2>8=2834� C74= C74 3D0;8CH 60? F8;; E0=8B7� 0=3 C74 B>;DC8>=B >5 ��0=3 �� F8;; 14 270A02C4A8I43 1H C74 2><?;4<4=C0AH B;02:=4BB 2>=38C8>=B ��=34AB>=0=3 "0B7� �� B 8= C74 ?A>>5 >5 '74>A4< 4� C74 2><?;4<4=C0AH B;02:=4BB 2>=38�C8>=B 14CF44= �� 0=3 �� 2>8=2834 F8C7 �� B> 8C >=;H A4<08=B C> ?A>E4 C70C ��02784E4B 8CB >?C8<0; E0;D4 0=3 8C 2>8=2834B F8C7 C74 ��>?C8<0; E0;D4� �>=B834A 0B>;DC8>= x� >5 �� 0=3 0 B>;DC8>= (f �, z�, ��) >5 �� F7>B4 4G8BC4=24 8B 6D0A0=C4431H '74>A4< 4a�� >=B834A

��d = J��z� + wf � +

�E� + w

��+

0=3

��u = J��z� + wf � +

�E� + w

��.

Obviously, (f !, z!, !!,"!d,"!u) is (33)-feasible, and the dual objective becomes

! H,"!d " # ! 1,"!u " #

(E! + w

)! 1, J!(!!) "

=! H,MaxnJ!(z! + wf ! +

(E! + w

)!!), 0o"

# ! 1,Maxn#J!

(z! + wf ! +

(E! + w

)!!), 0o+(E! + w

)J!(!!) ",

which coincides with the (25)-optimal value because there is no duality gap between(25) and (26) (Theorem 4). Hence, since the (25)-optimal value is an upper valuefor the (33)-objective (Anderson and Nash, 1987), (f !, z!, !!,"!d,"

!u) solves (33) and

there is no duality gap between (25) and (33). !

Remark 5. Theorem 5 implies that for appropriate PU , # and ! the dual solution(f !, z!, !!,"!d,"

!u) of (33) may be obtained in practice by linear programming methods,

which is a very good new because interesting algorithms to solve these problems areavailable in both, Önite and inÖnite dimensions (Anderson and Nash, 1987). Oncethe dual solution was computed, the optimal retention J(x!) may be obtained by(34), which is much easier to apply once we know (f !, z!, !!,"!d,"

!u). Besides, if (25)

has a linear dual, then this problem should be ìalmost linearîtoo. Sections 4 and5 will show two di§erent methods linearizing Problem (25) (see Problems (36) and(40) below). !

4. On the existence of bang-bang solutions

The literature about the optimal reinsurance problem frequently obtains stop-loss orclosely related solutions such as stop-loss contracts with an upper bound, given by

J (1# x!) (t) = (t# a)+ # (t# b)+ ,

0 $ t $ M and 0 $ a b0, if a < t < b

and Lemma 6 below will show that this is an extreme point of the (25)-feasible setif H = 0. A natural question is how general this result is. Let us prove Theorem7 below showing that some extreme points are ìgood approximationsîof the (25)-solution for general choices of H and general uncertainty levels for both the insurerand the reinsurer.

2Balb·s et al. (2009) and (2011) obtain stop-loss optimal contracts even for many problemswithout comonotonicity restrictions.

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

�� %=887;- <0)< x 1; � ��.-);1*4-� &0-6� <0- +76,1<176; *-47? ):- -9=1>)�4-6<�a) x 1; ) -@<:-5- 8716< 7. <0- � ��.-);1*4- ;-< H � x � 1�b) &0-:- -@1;< ) �7:-4�5-);=:)*4- 8):<1<176 [0,M ] = A�B ;=+0 <0)< x (t) = H (t)

1. t � A )6, x (t) = 1 1. t � B��-6+-.7:<0� <0- .-);1*4- ;74=<176 x )*7>- ?144 *- +)44-, *)6/�*)6/ :-<-6<176�

�#!!�� b) =� a) 8B >1E8>DB� B> ;4C DB ?A>E4 C74 2>=E4AB4 8<?;820C8>=� &D??>B4 C70Ca) 7>;3B� �>=B834A C = {0 � t �M ;H (t) < x (t) < 1} 0=3

Cn =

�0 � t �M ;H (t) +

1

n< x (t) < 1�

1

n

�

5>A n � �� #1E8>DB;H� C = ��n=1Cn� �5 L 8B C74 414B6D4 <40BDA4 C74= L (C) =LimnL (Cn) 1420DB4 (Cn)n�IN 8B 0= =>= 342A40B8=6 B4@D4=24� �4=24� L (C) = 0 85L (Cn) = 0 5>A 4E4AH n� �5 5>A B><4 n F4 70E4 L (Cn) > 0 C74= H (t) < x (t)�

1n� 1

0=3 H (t) � x (t) + 1n< 1 5>A t � Cn� B> 5D=2C8>=B

x1 (t) = x (t)�1n, x2 (t) = x (t) +

1n

5>A t � Cn 0=3 x1 (t) = x2 (t) = x (t) 5>A t /� Cn 0A4 1>C7 ��540B81;4� '7DB�x = 1

2x1 +

12x2 2>=CA0382CB a)� �

��!#�� %=887;- <0)< E� + w � 0� �. x� ;74>-; � � <0-6 .7: ->-:A n � ��

<0-:- -@1;<; ) *)6/�*)6/ .-);1*4- :-<-6<176 x�n ;=+0 <0)< <0- 7*2-+<1>- .=6+<176 7. � �;)<1;C-;

(�J (x�)) + w (��U (�J (x�))) +

�E� + w

�� (J (1� x�))

� (�J (x�n)) + w (��U (�J (x�n))) +

�E� + w

�� (J (1� x�n)) +

1n.

��

�#!!�� 4C (f �, z�, ��) B>;E4 �� '74=� �� 0=3 �� B7>F C70C

(�J (x�)) = ��IP� (J (x�) z�)

��U (�J (x�)) = ��IP� (J (x

�) f �)� (J (1� x�)) = ��IP� (J (1� x

�) ��) .

'74 ��540B81;4 B4C 8B F40:;H�2><?02C� 0=3 C74A45>A4� 022>A38=6 C> C74 �A48=�!8;<0='74>A4<� 85 n � �� C74= C74 ;8=40A >?C8<8I0C8>= ?A>1;4<

Min ��IP� (J (x) z

�) + w (��IP� (J (x) f�))�

�E� + w

��IP� (J (1� x) �

�)

H � x � 1��

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

B0C8BJ4B C74 4G8BC4=24 >5 0= 4GCA4<4 ?>8=C �i.e.� 0 10=6�10=6 A4C4=C8>=� B44 4<<0 6�H � x�n � 1 BD27 C70C

��IP� (J (x�) z�) + w (��IP� (J (x

�) f �))��E� + w

��IP� (J (1� x

�) ��) �

��IP� (J (x�n) z

�) + w (��IP� (J (x�n) f

�))��E� + w

��IP� (J (1� x

�n) �

�) + 1n

��=34AB>= 0=3 "0B7� �� 4=24� �� 0=3 �� CA8E80;;H ;403 C>

��IP� (J (x�n) z

�) + w (��IP� (J (x�n) f

�)) +�E� + w

��IP� (J (1� x

�n) (��

�))

� (�J (x�n)) + w (��U (�J (x�n))) +

�E� + w

�� (J (1� x�n))

0=3 �� 142><4B >1E8>DB� �

��#� � �76,1<176 E� + w � 0 ?144 .:-9=-6<4A 074,� �7: 16;<)6+-� )++7:,16/ <7$-5):3 3� 1< 074,; .7: -@8-+<)<176 *7=6,-, :1;3 5-);=:-;� ,->1)<176 5-);=:-;� )6,<0-1: :7*=;< -@<-6;176;�&0-7:-5 7 /=):)6<--; <0)< )88:78:1)<- *)6/�*)6/ :-<-6<176; ?144 *- ); +47;- );

,-;1:-, <7 <0- 78<15)4 ;74=<176� &0- -@1;<-6+- 7. ) *)6/�*)6/ 78<15)4 :-<-6<176 +)6 *-8:7>-, 1. <0- 416-): 8:7*4-5 �� 0); )6 -@<:-5- ;74=<176� �6,-:;76 )6, );0 ��;07? <0)< <01; -@<:-5- ;74=<176 +)6 *- /=):)6<--, =6,-: ),-9=)<- );;=58<176;� �7:-@)584-� 1. � � �7: �� 1; ) C61<-�,15-6;176)4 8:7*4-5� ?01+0 074,;� .7: 16;<)6+-�1. <0- <7<)4 5);; 7. ��0 1; +76+-6<:)<-, 76 ) C61<- ;=*;-< 7. [0,M ]� �

��#� � � =5-:1+)4 -@8-:15-6<;�� 76,1<176; �� 7. &0-7:-5 4 ?-:- =;-.=4 <78:7>- <0- -@1;<-6+- 7. *)6/�*)6/ ;74=<176;� )6, <0-A ?144 *- )4;7 =;-.=4 <7 16 7:,-: <7;-- <0)< ?- +)667< /7 *-A76, &0-7:-5 7 )6, /1>- +47;-, -@8:-;;176; .7: ;-<;

A = {t;x� (t) = H (t)} and B = {t;x� (t) = 1} ,

J(x�) ,-67<16/ <0- 78<15)4 +76<:)+<� '6,-: 57:- +4);;1+)4 )88:7)+0-; J(x�) 1; );<78�47;; :-16;=:)6+- 7: ) +47;-4A :-4)<-, 76-� �7: ) ;<78�47;; +76<:)+< <0-:- -@1;<;M0 � [0,M ] ;=+0 <0)<

A = (M0,M) and B = (0,M0) .

->-:<0-4-;;� 16 7=: ;-<<16/ A )6, B +:1<1+)44A ,-8-6, 76 <0- )5*1/=1<A ;-< PU )6,<0- :1;3 .=6+<176; )6, �� )6, 5)6A ,1F-:-6< ;1<=)<176; 5)A ):1;-� �-< =; ;07? <01;.)+< ?1<0 ;1584- 144=;<:)<1>- -@)584-;� 7<1+- <0)< <0-;- -@)584-; )4;7 )884A 16 <0-676�)5*1/=7=; )6, 416-): +);-�%=887;- <0)< M = 5� <0- ;=887:< 7. ��0 1; {1, 2, 3, 4, 5} )6, ��0 (i) = 0.2� i =

1, 2, 3, 4, 5� %=887;- <0)< R = 1 �676�)5*1/=7=; +);-� )6, <0-:-.7:- PU 764A +76<)16;

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 ��

<0- B-:7�>):1)6+- :)6,75 >):1)*4- f = 1� %=887;- C6)44A <0)< )6, � ):- 416-):� ��

764A +76<)16; <0- -4-5-6< z = 1 �;7 E� = 1� )6, �� 764A +76<)16; � = �c � L2 (��0)

?1<0 c � 0 �;7 � 1; /1>-6 *A �� -67<- ci = c (i) � 0� i = 1, 2, 3, 4, 5� )6, <)3-w = 3 16 � ��%16+- E� = 1 )6, ��0 (z

� + wf � = 4) = 1� �@8:-;;176 �� 15841-; <0)<�

��

��

s < 1� J��E� + w

�c�= 0.6

�5i=1 ci,

1 < s < 2� J��E� + w

�c�= 0.6

�5i=2 ci.

s < 2 < 3� J��E� + w

�c�= 0.6

�5i=3 ci,

s < 3 < 4� J��E� + w

�c�= 0.6 (c4 + c5) ,

4 < s� J��E� + w

�c�= 0.6c5,

J� (z� + wf �) = 12

J� (z� + wf �) = 9.6

J� (z� + wf �) = 7.2

J� (z� + wf �) = 4.8

J� (z� + wf �) = 2.4

&0=;� �� 4-),; <7 <0- *)6/�*)6/ 78<15)4 ;74=<176 x� (t) +0):)+<-:1B-, *A

��

��

t < 1, 0.6�5

i=1 ci < 12� x� = H

t < 1, 0.6�5

i=1 ci > 12� x� = 1

1 < t < 2, 0.6�5

i=2 ci < 9.6� x� = H

1 < t < 2, 0.6�5

i=2 ci > 9.6� x� = 1

2 < t < 3, 0.6�5

i=3 ci < 7.2� x� = H

2 < t < 3, 0.6�5

i=3 ci > 7.2� x� = 1

3 < t < 4, 0.6 (c4 + c5) < 4.8� x� = H

3 < t < 4, 0.6 (c4 + c5) > 4.8� x� = 1

4 < t, 0.6c5 < 2.4� x� = H

4 < t, 0.6c5 > 2.4� x� = 1

��

%=887;- <0)< ci = 6� i = 1, 2, 3 )6, ci = 2.5� i = 4, 5� &0-6� �� 4-),; <7 x� = 1 1.

t < 2 )6, x� = H 1. t > 2 ?01+0 *-+75-; <0- ;<78�47;; +76<:)+< J(1� x�) = (t� 2)+

16 <0- +4);;1+ ;+-6):17 H = 0� ->-:<0-4-;;� <0- 7887;1<- ;1<=)<176 5)A )4;7 ;74>-� �� 6 .)+<� 1. <0- :-16;=:-: =;-; ) >):1)*4- 47),16/ :)<- � (t) 8-6)41B16/ 01/0 +4)15;;=+0 ); ci = 1.05� i = 1, 2, 3� )6, ci = 5� i = 4, 5� <0-6 �� 4-),; <7 x� = H 1.t < 3 )6, x� = 1 1. t > 3 ?01+0 *-+75-; J(x�) = (t� 3)+ 16 <0- +4);;1+ +);- H = 0�

�=>C824 C70C J��E� + w

�c�(i) 0=3 J� (z� + wf�) (i) 0A4 =>C A4;4E0=C 5>A i = 1, 2, 3, 4, 5� B8=24

C74 414B6D4 <40BDA4 >5 C78B B4C E0=8B74B 0=3 J��E� + w

�c�, J� (z� + wf�) � L2 [0,M ]�

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 ��

�16)44A� 57:- +7584-@ *)6/�*)6/ ;74=<176; ):- )4;7 .-);1*4-� �7: 16;<)6+-� 1. c1 = 7�c2 = 1� c3 = 6� c4 = 1 )6, c5 = 6� <0-6 x

� = H .7: t � (1, 2) � (3, 4) )6, x� = 1 .7:t � (0, 1) � (2, 3) � (3, 5)� �

5. T"� )*.#'�& ,�#(-/,�(�� &#(��, *,)�&�'

�B 8=3820C43 8= %4<0A: 5�'74>A4< 5 8<?;84B C70C 5>A 0??A>?A80C4 PU � 0=3� C74 3D0;B>;DC8>= (f �, z�, ��,��d,�

�u) >5 �� <0H 14 >1C08=43 8= ?A02C824 1H ;8=40A ?A>6A0<<8=6

<4C7>3B� #=24 (f �, z�, ��,��d,��u) 8B :=>F=� C74 ;8=40A ?A>1;4< �� <0H ;403 C> C74

>?C8<0; A4C4=C8>= J (x�) �B44 C74 ?A>>5 >5 '74>A4< 7�� "4E4AC74;4BB� C74A4 8B B42>=3<4C7>3 0;;>F8=6 DB C> ;8=40A8I4 �� 0=3 C78B 8B C74 5>2DB >5 C78B B42C8>=�

&D??>B4 C70C C74 BD1�6A0384=CB >5 0=3 � 0A4 68E4= 1H ;8=40A 2>=BCA08=CB �A>1DBCCV aR� A>1DBC 01B>;DC4 34E80C8>=� <0=H 20B4B >5 C74 A>1DBC F4867C43 CV aR� 4C2�� B44'74>A4< 2 0=3 %4<0A: 3�� '74=� F4 20= 68E4 0 ;8=40A 3D0; >5 �� =34AB>= 0=3"0B7� �� F7827 8B C74 3>D1;4�3D0; >5 �� 0=3 2>D;3 ;403 C> C74 >?C8<0; A4C4=C8>=C>>� �= >C74A F>A3B� C74 >?C8<0; A4C4=C8>= 2>D;3 14 38A42C;H >1C08=43 1H B>;E8=6 0;8=40A ?A>1;4<� �= >A34A C> B7>AC4= C74 4G?>B8C8>= ;4C DB 5>2DB >= C74 A>1DBC CV aR�1DC C74B4 8340B 0??;H 8= <D27 <>A4 5A0<4F>A:B�

&D??>B4 C70C = RCV aR(�U ,�)0=3 � (y) = ��IP� (y) + �RCV aR(�U ,�)

(�y)�

�, � � 0 �B44 �� F74A4 C74 D=24AC08=CH ;4E4; >5 C74 A48=BDA4A P0U 8B 64=4A0C43 1HC74 B4C >5 ?A8>AB

PU =�h � L2 (��0) ; 0 � h � R, ��IP� (h) = 1

�, ��

F74A4 R � L2 (��0) 0=3 R � 1� �B 0;A403H B083� 85 R � R C74= C74 A48=BDA4AD=24AC08=CH F8;; 14 78674A C70= C74 8=BDA4A >=4� '78B 8B 0 @D8C4 =0CDA0; 0BBD<?C8>=1420DB4 C74 C74 A48=BDA4A 8=5>A<0C8>= 01>DC C74 8=E>;E43 ?>;8284B <0H 14 ;>F4A� 1DCF4 F8;; =>C 8<?>B4 8C� �= 64=4A0;� F4 F8;; =>C =443 0=H A4;0C8>=B78? 14CF44= R 0=3R� &8<8;0A;H� F4 F8;; =>C 8<?>B4 0=H A4;0C8>=B78? 14CF44= C74 2>=J34=24 ;4E4;B � 0=3�� DAC74A<>A4� 5>A R = 1 �R = 1� F4 8=2;D34 C74 =>=�0<186D>DB 8=BDA4A �A48=BDA4A�8= C74 0=0;HB8B� 0=3� 0=0;>6>DB;H� 5>A R = 1� � � 1 0=3 � = 0 F4 8=2;D34 C74 EV PP0B 0 ?0AC82D;0A 20B4�

'0:4 C = (1 + w)��IP� (J (1))� $A>?>B8C8>= 1� �G?A4BB8>= �� '74>A4< 2� �G�

">C824 C70C C74 B>;DC8>= >5 C78B 4G0<?;4 8B 8= C74 ;8=4 >5 C7>B4 B>;DC8>=B 5>D=3 1H �D8 et al. ��5>A A8B: 5D=2C8>=B 0=3 ?A4<8D< ?A8=28?;4B 68E4= 1H 38BC>AC8>=B� '74 8=C4A?A4C0C8>= >5 '74>A4< 78B C70C C78B CH?4 >5 64=4A0; B>;DC8>= BC8;; 0??;84B 8= 0<186D>DB 5A0<4F>A:B F8C7 A8B: 5D=2C8>=B 0=3?A4<8D< ?A8=28?;4B 8=2>A?>A0C8=6 1>C7 A8B: 0=3 0<186D8CH 0E4AB8>=� 0B F4;; 0B 8<?>B8=6 0 ;>F4A1>D=3 H F7827 C>C0;;H 4;8<8=0C4B C74 A48=BDA4A <>A0; 70I0A3�

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

?A4BB8>=B �� 0=3 �� 0=3 '74>A4< 5 8<?;H C70C �� 142><4B��

��

Max H,�d � 1,�u +(1 + w) � 1, J�(�) +C�

��

��

�J� (z + wf � (1 + w) ��)� �d + �u= (1 + w)�J� (1)

��IP� (f) = 1

f � R

��IP� (z) = 1

z � g1��

��

��

��IP� (g) = 1g � R

��IP� (�) = 1� � h

1��

��IP� (h) = 1

��

�

h � R�d,�u � L

2 [0,M ] , f, z, g, �, h � L2 (��0)�d,�u, f, z, g, �, h � 0

��

&8=24 C78B ?A>1;4< 8B >1E8>DB;H ;8=40A� F4 20= 2>=BCAD2C 8CB ;8=40A 3D0; � D4=14A64A�� >A �=34AB>= 0=3 "0B7� �� F7827 F8;; 14 20;;43 3>D1;4�3D0; >5 �� 0=3 8B68E4= 1H

Min

C � (1 + w)��IP� (J (�1)) + �2 + ��IP� (R�3) + �4

+�6 + ��IP� (R�7) + �8 + �10 + ��IP�

�R�11

�

s.t.

��

��

H � �1 � 0�1 � 1 � 0wJ (�1)� �2 � �3 � 0J (�1)� �4 � �5 � 011��

�5 � �6 � �7 � 0

��

��

(1 + w) �J (1� �1)� �8 � �9 � 011��

�9 � �10 � �11 � 0

�1 � L2 [0,M ] , �2,�4,�6,�8,�10 � ��

�3,�5,�7,�9,�11 � L2 (��0)

�3,�5,�7,�9,�11 � 0

� �

'74 2><?;4<4=C0AH B;02:=4BB 2>=38C8>=B 14CF44= �� 0=3 � � 0A4��

��

��

��

��d (��1 �H) = 0

��u (1� ��1) = 0f � (��2 + ��3 � wJ (�

�1)) = 0

z� (��4 + ��5 � J (��1)) = 0

g��6 + ��7 �

11��

��5

�= 0

��

��

�� (��8 + ��9 � (1 + w) �J (1� ��1)) = 0

h��10 + ��11 �

11��

��9

�= 0

��3 (f� �R) = 0

��5

�z� � g�

1��

�= 0

��

��

��7 (g� �R) = 0

��9

�� h�

1��

�= 0

��11

�h� � R

�= 0

� �

'74 2>=BCA08=CB >5 �� 0=3 � �� 0;>=6 F8C7 � �� 270A02C4A8I4 C74 3D0; 0=3 C743>D1;4�3D0; B>;DC8>=B

D = (��d,��u, f

�, z�, g�, ��, h�) � ��

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 ��

0=3S = (��1,�

�2,�

�3,�

�4,�

�5,�

�6,�

�7,�

�8,�

�9,�

�10,�

�11) � ��

85 � � 8B B>;E01;4 0=3 C74A4 8B => 3D0;8CH 60? 14CF44= �� 0=3 � � ��=34AB>= 0=3"0B7� �� >=E4AB4;H� 85 C74A4 4G8BC ��540B81;4 0=3 � ��540B81;4 B>;DC8>=B >5 � �C74= � � 8B B>;E01;4 0=3 C74A4 8B => 3D0;8CH 60? 14CF44= �� 0=3 � �� = 64=4A0;�C74 B>;E018;8CH >5 � � 0=3 C74 01B4=24 >5 3D0;8CH 60? 20= =>C 14 6D0A0=C443�

"443;4BB C> B0H� 8C 8B 8<?>AC0=C C> B44 C70C C74 B>;DC8>= >5 � � ;403B C> C74 >?C8<0;B>;DC8>= x� >5 �� 0=3 C74 >?C8<0; A4C4=C8>= J (x�)� $A>?>B8C8>= 8 14;>F B7>FB C70CC74 01B4=24 >5 3D0;8CH 60? ?;0HB 0 2A8C820; A>;4�

�#!"!$�%�! �� %=887;- <0)< <0- ,7=*4-�,=)4 8:7*4-5 �� 1; ;74>)*4- )6, 0); 67,=)41<A /)8 ?1<0 <0- ,=)4 8:7*4-5 �� %=887;- <0)< �� ;74>-; <0- ,7=*4-�,=)4�� &0-6� x� = ��1 ;74>-; <0- 78<15)4 :-16;=:)6+- 8:7*4-5 � ��

�#!!�� >=B834A C74 B>;DC8>= � �� >5 �� F7>B4 4G8BC4=24 8B 6D0A0=C443 1H '74�>A4< 5� �22>A38=6 C> '74>A4< 5 F4 <DBC ?A>E4 C70C �� 7>;3B 5>A ��1 0=3 � ��'74 ?A>?>B8C8>= 0BBD<?C8>=B 8<?;H C70C � � 7>;3B 0=3 C74A45>A4 C74 5>DAC7 0=3 J5C72>=38C8>=B 8= �� 7>;3 C>>� '7DB� 8C >=;H A4<08=B C> B44 C70C C74 JABC� B42>=3 0=3C78A3 2>=38C8>=B 8= �� 7>;3� &8=24 C74 C7A44 8=4@D0;8C84B 70E4 0 B8<8;0A ?A>>5� ;4C DBB7>F C70C

��IP� (J (��1) z

�) � ��IP� (J (��1) z) � �

7>;3B 5>A 4E4AH z � �� 0=3 ;4C DB ><8C C74 CF> A4<08=8=6 20B4B� '74 5>DAC7 4@D0C8>=>5 � � 8<?;84B C70C J (��1) z

� = z� (��4 + ��5)� '74 =8=C7 4@D0C8>= � � 8<?;84B C70CJ (��1) z

� = z��4 +g�

1��5� '74 J5C7 4@D0C8>= ;403B C> J (�

�1) z

� = z��4 + g��6 + g

��7�

0=3 C74 C4=C7 >=4 J=0;;H ;403B C> J (��1) z� = z��4 + g

��6 + R��7� �4=24� B8=24

��IP� (z�) = ��IP� (g

�) = 1�

��IP� (J (��1) z

�) = ��4 + ��6 + ��IP� (R��7) . � ��

&D??>B4 C70C z � �� '74 2>=BCA08=CB >5 �� 8<?;H C74 4G8BC4=24 >5 g � PU �B44� �� BD27 C70C z � g

1�� 4=24� J (��1) z � z (��4 + ��5) 3D4 C> z � 0 0=3 C74 5>DAC7

2>=BCA08=C >5 � �� J (��1) z � z��4 +g1��

��5 3D4 C> z �g1��

0=3 ��5 � 0� J (��1) z �

z��4 + g��6 + g�

�7 3D4 C> g � 0 0=3 C74 J5C7 2>=BCA08=C >5 � �� 8=0;;H� J (��1) z �

z��4 + g��6 +R�

�7 3D4 C> g � R 0=3 ��7 � 0� �4=24� ��IP� (z) = ��IP� (g) = 1 ;403B C>

��IP� (J (��1) z) � ��4 + ��6 + ��IP� (R�

�7) . � ��

� �� 0=3 � �� 8<?;H � ��

$A>?>B8C8>= 8 8B DB45D; 8= ?A02C824 1420DB4 8C <0:4B C74 >?C8<0; A48=BDA0=24 ?A>1�;4< 142><4 ;8=40A� 1DC C74 01B4=24 >5 3D0;8CH 60? 14CF44= �� 0=3 � � <DBC 14

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 ��

6D0A0=C443� '74>A4< 10 14;>F F8;; B7>F C70C C78B 8B C74 20B4 85 H� R 0=3 R 0A4B8<?;4 5D=2C8>=B� %420;; C70C B8<?;4 5D=2C8>=B 0A4 68E4= 1H C74 64=4A0; 4G?A4BB8>=�n

i=1 ai�Bi� [0,M ] = B1 � B2 � ... � Bn 148=6 0 <40BDA01;4 0=3 38B9>8=C ?0AC8C8>=�a1, a2, ....an 148=6 A40; =D<14AB� 0=3 �Bi 148=6 C74 8=3820C>A >5 Bi� i = 1, 2, ..., n� #1�

E8>DB;H� 2>=BC0=C 5D=2C8>=B �BD27 0B C74 DBD0; >=4B H = 0� R = R = 1� 0A4 8=2;D343�!>A4>E4A� A420;; C74 C74 B4C >5 B8<?;4 5D=2C8>=B 8B 34=B4 8= L2 [0,M ] 0=3 L2 (��0)� 0=35>A 64=4A0; H� R 0=3 R >=4 20= 0;F0HB 1D8;3 M6>>3 B8<?;4 0??A>G8<0C8>=BN�

�= >A34A C> ?A>E4 '74>A4< 10 F4 F8;; =443 B><4 2>=24?CB 0=3 8=BCAD<4=C0; A4BD;CBF7827 F8;; 14 1A84KH BD<<0A8I43� �5 B0 � B 8B 0 BD1��0;641A0 >5 C74 �>A4; �0;641A0B >5 [0,M ]� C74=� F8C7 >1E8>DB =>C0C8>=B� L2 [0,M,B0] 8B 0 2;>B43 BD1B?024 >5 L

2 [0,M ]0=3 L2 (��0,B0) 8B 0 2;>B43 BD1B?024 >5 L

2 (��0)� '74 >AC7>6>=0; ?A>942C8>= >5 L2 (��0)

>E4A L2 (��0,B0) 8B 68E4= 1H C74 2>=38C8>=0; 4G?42C0C8>=

L2 (��0) � L2 (��0,B0)y � ��IP� (y |B0 )

0=3 8C 8B :=>F= C70C��IP� (y1y2) = ��IP� (��IP� (y1 |B0 ) y2) � ��

5>A 4E4AH y1 � L2 (��0) 0=3 4E4AH y2 � L2 (��0,B0)� �4B834B� 85 L 8B C74 414B6D4<40BDA4 C74=

M8B 0 ?A>1018;8CH� C74 >AC7>6>=0; ?A>942C8>=

L2 [0,M ] � L2 [0,M,B0]y � �� L

M(y |B0 )

8B 0;B> 68E4= 1H C74 1H 0 2>=38C8>=0; 4G?42C0C8>=� 0=3 � �� 142><4B

�� L

M(y1y2) = �� L

M

�� L

M(y1 |B0 ) y2

�. � ��

�5 J0 8B C74 2><?>B8C8>= >5 J 0=3 C74 >AC7>6>=0; ?A>942C8>=B 01>E4�

J0 := ��IP� (� |B0 ) � J � �� L

M(� |B0 ) ,

C74=� 140A8=6 8= <8=3 C70C 4E4AH >AC7>6>=0; ?A>942C8>= 8B B4;5�039>8=C� >=4 70B C70C

J�0 =��IP� (� |B0 ) � J � �� L

M(� |B0 )

��

= �� L

M(� |B0 )

� � J� � ��IP� (� |B0 )�

= �� L

M(� |B0 ) � J

� � ��IP� (� |B0 ) .

�= >C74A F>A3B� 85 J0 : L2 [0,M ]� L2 (��0) 8B 68E4= 1H

J0 (y) := ��IP�

�J�� L

M(y |B0 )

�|B0

��

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 ��

C74= J�0 : L2 (��0)� L2 [0,M ] 8B 68E4= 1H

J�0 (y) = �� L

M(J� (��IP� (y |B0 )) |B0 ) . ��

�� -< (Bn)�

n=1 *- ) C4<:)<176 7. ;=*��)4/-*:); 7. B�a) �-< (un)

�

n=1 � L2 (��0) �:-;8-+<1>-4A� (un)�

n=1 � L2 [0,M ]� *- ) 5):<16/)4-),)8<-, <7 <0- C4<:)<176 (Bn)

�

n=1� �. (un)�

n=1 1; 67:5 *7=6,-,� <0-6 <0-:- -@1;<; u �L2 (��0) �:-;8-+<1>-4A� u � L

2 [0,M ]� ;=+0 <0)< (un)�

n=1 +76>-:/-; <7 u *7<0 16 67:5)6, )457;< ->-:A?0-:-� �7:-7>-:� un = ��IP� (u |Bn ) �:-;8-+<1>-4A� un = �� L

M(u |Bn )�

.7: ->-:A n � ��b) �. <0-:- -@1;<; u � L2 (��0) �:-;8-+<1>-4A u � L

2 [0,M ]� ;=+0 <0)< un = ��IP� (u |Bn )�:-;8-+<1>-4A� un = �� L

M(u |Bn )� )6, B = (��n=1Bn) +716+1,-; ?1<0 <0- �)4/-*:)

/-6-:)<-, *A ��n=1Bn� <0-6 (un)�

n=1 +76>-:/-; <7 u *7<0 16 67:5 )6, )457;< ->-:A�?0-:-�c) %=887;- <0)-:/-; <7 J�

=61.7:54A 16 A�

�#!!�� a) 0=3 b) 0A4 2>=B4@D4=24B >5 C74 �>>1LB !0AC8=60;4 �>=E4A64=24 '74>A4<��DAA4CC� �� B> ;4C DB ?A>E4 c)� �BB4AC8>= b) 8<?;84B C70C A � L2 (��0) 8B 0 2><?02CB4C� �4B834B ��IP� (� |Bn ) � J

�� IP� (� |Bn )

�� J� � �J�

8<?;84B C70C�� L

M(� |Bn ) � J

��n=1

8B 1>D=343 0=3 C74A45>A4 4@D8�2>=C8=D>DB� '7DB�

C74 �B2>;8LB '74>A4< 8<?;84B C70C�� L

M(� |Bn ) � J

��n=1

2>=E4A64B C> J� D=85>A<;H 8=

A 85�� L

M(J�(v) |Bn )

��n=1

2>=E4A64B C> J�(v) 5>A 4E4AH v � A� F7827 7>;3B 3D4 C> b)

��84D3>==S� ��

��!#�� %=887;- <0)< � > 0 )6, � > 0� �. H� R )6, R ):- ;1584- .=6+<176;<0-6 <0- ,7=*4-�,=)4 8:7*4-5 �� 1; ;74>)*4- )6, 0); 67 ,=)41<A /)8 ?1<0 <0- ,=)48:7*4-5 ��

�#!!�� C 8B :=>F= C70= C74 C>?>;>6H >5 [0,M ] 8B 64=4A0C43 1H 0 2>D=C01;4 10B8B >5A4;0C8E4 >?4= B4CB ��4;;H� �� B> C74 �>A4; �0;641A0 B 8B 64=4A0C43 1H 0 2>D=C01;4B4@D4=24 (Bn)

�

n=1 � B� �>=B834A C74 �0;641A0 Bn = (B1, B2, ..., Bn) 5>A 4E4AHn � �� C 8B >1E8>DB C70C 4E4AH Bn 70B 0 J=8C4 =D<14A >5 4;4<4=CB� 0=3 C74 J;CA0C8>=(Bn)

�

n=1 B0C8BJ4BB = (��n=1Bn) . ��

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 ��

&8=24 H� R 0=3 R 0A4 B8<?;4 5D=2C8>=B F4 70E4 C70C C74H 64=4A0C4 0 �0;641A0

�H,R, R

�F8C7 J=8C4;H <0=H <40BDA01;4 B4CB� '7DB� 1H A4?;028=6 Bn F8C7

�Bn �

�H,R, R

��,

F4 >1C08= 0 =4F J;CA0C8>=� BC8;; 34=>C43 1H (Bn)�

n=1� F7827 0;B> B0C8BJ4B C70C 4E4AH�0;641A0 8= C74 J;CA0C8>= 70B J=8C4;H <0=H 4;4<4=CB� !>A4>E4A� �� BC8;; 7>;3B� 0=3H� R 0=3 R 0A4 Bn�<40BDA01;4 5>A 4E4AH n � ��

�>=B834A C74 ;8=40A >?C8<8I0C8>= ?A>1;4< ��n� �A4B?42C8E4;H� � �n�� B8<8;0AC> �� A4B?42C8E4;H� � �� 85 >=4 A4?;024B �B44 � �� 0=3 �� J � J�� L2 [0,M ] 0=3L2 (��0) F8C7 Jn = ��IP� (� |Bn ) �J � �� L

M(� |Bn )� J

�n = �� L

M(� |Bn ) �J

� � ��IP� (� |Bn )�

L2 [0,M,Bn] 0=3 L2 (��0,Bn)� �C 8B 40BH C> B44 C70C � �n� 8B C74 3D0; ?A>1;4< >5

��n�� DAC74A<>A4� B8=24 1>C7 ?A>1;41E8>DB;H 70E4 540B81;4 B>;DC8>=B� C74H0A4 1>D=343� �8=0;;H� B8=24 Bn >=;H 70B J=8C4;H <0=H 4;4<4=CB� ��n� 0=3 � �n�0A4 J=8C4�38<4=B8>=0; ?A>1;4=0;B?024B�� B> C74H 0A4 B>;E01;4 0=3 C748A >?C8<0; E0;D4B 2>8=2834 5>A 4E4AH n � ��4=>C4 1H n C74 >?C8<0; E0;D4 >5 ��n� 0=3 � �n�� 1H

Dn = (��d,n,�

�u,n, f

�n, z

�n, g

�n, �

�n, h

�n) ��

0 B>;DC8>= >5 ��n� 0=3 1H

Sn = (��1,n,�

�2,n,�

�3,n,�

�4,n,�

�5,n,�

�6,n,�

�7,n,�

�8,n,�

�9,n,�

�10,n,�

�11,n)

0 B>;DC8>= >5 � �n�� '74H B0C8B5H C74 2>=BCA08=CB >5 ��n� 0=3 � �n�� F7827 0A4C7>B4 >5 �� 0=3 � � F8C7 C74 >1E8>DB <>38J20C8>=B� 0=3 C74H 0;B> B0C8B5H � �n��F7827 2>8=2834B F8C7 � � D=34A BCA0867C5>AF0A3 <>38J20C8>=B C>>� �>=B834A C74B>;DC8>= D �B44 � �� >5 �� 0=3 34=>C4 1H C74 >?C8<0; E0;D4 >5 C78B ?A>1;4< �C74H4G8BC >F8=6 C> '74>A4A 4E4AH n � �� =3443� ;4C DB B44 C70C n � � �� 8B 2;40A;H

��540B81;4 85 (��d,n,��u,n) 8B A4?;0243 1H

�d = (J� (z� + wf � � (1 + w) (�� )))+

�u = (J� (z� + wf � � (1 + w) (�� )))� .

&8=24 H 8B Bn�<40BDA01;4 0=3 y � � L

M(y |Bn ) 8B 8=2A40B8=6� � �� ;403B C>

H,�d = H, �� L

M(�d |Bn ) =

H, �� L

M

�(J� (z�n + wf

�n � (1 + w) (��

�n � �)))+ |Bn

�

� H,�� L

M(J� (z�n + wf

�n � (1 + w) (��

�n � �)) |Bn )

�+= H,��d,n .

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 ��

&8<8;0A;H� � 1,�u � � 1,��u,n � 0=3 � �� 8<?;84B C70C

1, J� (��) = 1, �� L

M(J� (��) |Bn ) .

�4=24� n � H,�d � 1,�u + 1, J� (��) � � '74 8=4@D0;8CH n � n+1

<0H 14 ?A>E43 F8C7 0 B8<8;0A <4C7>3�(ii) ( n)

�

n=1 2>=E4A64B C> � �=3443� 2>=B834A C74 B>;DC8>= � �� >5 �� 0=3 34=>C4

(f ��n , z��n , g

��n , �

��n , h

��n ) = ��IP� ((f

�, z�, g�, ��, h�) |Bn ) .

�5��d,n = (J

�n (z

��n + wf

��n � (1 + w) (��n � �)))+

��u,n = (J�n (z

��n + wf

��n � (1 + w) (��n � �)))�

C74=(��d,n,�

��u,n, f

��n , z

��n , g

��n , �

��n , h

��n )

8B ��n��540B81;4� B> C74 ��n��>1942C8E4 n 0C C78B ?>8=C B0C8BJ4B n � n� '74=�8C 8B BDP 284=C C> B44 C70C ( n)

�

n=1 2>=E4A64B C> � 4<<0 9 8<?;84B C70C

(f ��n , z��n , g

��n , �

��n , h

��n )

�n=1

2>=E4A64B C> (f �, z�, g�, ��, h�) 0=3�� L

M(� |Bn ) � J

��n=1

2>=E4A64B C> J� D=85>A<;H

>= (f ��n , z��n , g

��n , �

��n , h

��n )

�n=1� �4=24�

Limn��(��d,n) = Limn�� (J

�n (z

��n + wf

��n � (1 + w) (��n � �)))+

= Limn��

�� L

M((J� (z��n + wf

��n � (1 + w) (��n � �))) |Bn )

�+

((J� (z� + wf � � (1 + w) (�� ))))+ .

�=0;>6>DB;H�

Limn��(��u,n) = ((J

� (z� + wf � � (1 + w) (�� ))))� .

�>=B4@D4=C;H� 140A8=6 8= <8=3 C70C 1, J�n(��n ) = 1, J�(��n ) 2>=E4A64B C>

1, J�(��) �B44 � ��

Limn��( n) = Limn��

� H,��d,n � 1,�

��u,n +(1 + w) � 1, J

�n(�

��n ) +C

�

= H, ((J� (z� + wf � � (1 + w) (�� ))))+

� 1, ((J� (z� + wf � � (1 + w) (�� ))))�

+(1 + w) � 1, J�(��) +C = .

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

(iii) '74 B4@D4=24 (��1,n)�n=1 8B =>A<�1>D=343 1420DB4 C74 2>=BCA08=CB >5 � �n�

8<?;H C70C (��1,n)�n=1 � [H, 1]�

(iv) ��2,n � 0 5>A 4E4AH n � �� =3443� 85 ��2,n = �V < 0 C74 � �n��2>=BCA08=CB

F>D;3 ;403 C> 0 < V � wJ��1,n

�� 2,n � ��3,n� 0=3 � �n� F>D;3 8<?;H f

�n = R�

'7DB� R � 1 0=3 ��IP� (R) = ��IP� (f�n) = 1 F>D;3 8<?;H R = 1� �5 C78B 4@D0;8CH 3>4B

=>C 7>;3 C74= F4 F8;; 70E4 0 2;40A 2>=CA0382C8>=� �5 R = 1 7>;3B C74= 8C 8B 40BH C> B44C70C

(��1,n, 0,��3,n � V,�

�4,n,�

�5,n,�

�6,n,�

�7,n,�

�8,n,�

�9,n,�

�10,n,�

�11,n) ��

8B � �n��540B81;4 0=3 0=3 C74 >1942C8E4 5D=2C8>= A4<08=B C74 B0<4� i.e.� �� BC8;;B>;E4B � �n� 0=3 8CB B42>=3 2><?>=4=C 8B =>=�=460C8E4�(v) ��4,n � 0 0=3 ��8,n � 0 5>A 4E4AH n � �� =3443� 1>C7 20B4B 70E4 0 B8<8;0A

?A>>5� B> ;4C DB 340; F8C7 ��4,n� �5 ��4,n = �V < 0 C74 � �n��2>=BCA08=CB F>D;3 ;403

C> 0 < V � J��1,n

�� 4,n � ��5,n� 0=3 � �n� F>D;3 8<?;H z

�n =

g�n1��

� '74= 1 =

��IP� (z�n) = ��IP�

�g�n1��

�= 1

1��F>D;3 64=4A0C4 C74 2>=CA0382C8>= � = 0�

(vi) ��6,n � 0 0=3 ��10,n � 0 5>A 4E4AH n � �� =3443� 1>C7 20B4B 70E4 0 B8<8;0A?A>>5� B> ;4C DB 340; F8C7 ��6,n� �5 �

�6,n = �V < 0 C74 � �n��2>=BCA08=CB F>D;3 ;403

C> 0 < V � 11��

��5,n � ��6,n � ��7,n� 0=3 � �n� F>D;3 8<?;H g�n = R� '74 A4BC >5 C74

?A>>5 8B B8<8;0A C> (iv)�(vii) '74 B4@D4=24 (��2,n,�

�4,n,�

�6,n,�

�8,n,�

�10,n)

�n=1 8B 1>D=343� �=3443� 8C 8B BDP 284=C

C> B44 C70C C74A4 8B 0= D??4A 1>D=3� (i) B7>FB C70C

� C + (1 + w)��IP��Jn��1,n

��

2�

i=1

��IP�

�R��2i+1,n

�� IP�

�R��11,n

��

5�

i=1

��2i,n.

�4=24��2

i=1 ��IP�

�R��2i+1,n

�+ ��IP�

�R��11,n

�� 0 0=3 ��1,n � 1 ;403 C>

� C + (1 + w)��IP� (Jn (1)) = � C + (1 + w)��IP� (J (1)) �5�

i=1

��2i,n.

(viii) '74 B4@D4=24 (��3,n,��5,n,�

�7,n, ,�

�9,n,�

�11,n)

�n=1 8B =>A<�1>D=343� �=3443� C74

A4BCA82C8>=B 0=3 >1942C8E4 5D=2C8>= >5 � �n� 2;40A;H 8<?;H C70C

��3,n =�wJn

��1,n

�� 2,n

�+� wJn

��1,n

�� wJn (1)

��5,n =�Jn��1,n

�� 4,n

�+� Jn

��1,n

�� Jn (1)

��7,n =�

11��

��5,n � ��6,n

�+� 1

1��5,n �

11��Jn (1)

��9,n =�(1 + w) �Jn

�1� ��1,n

�� 8,n

�+� (1 + w) �Jn

�1� ��1,n

�� (1 + w) �Jn (1)

��11,n =�

11��

��9,n � ��10,n

�+� 1

1��9,n �

11��

(1 + w) �Jn (1)

O*.#'�& ,�#(-/,�(�� /(��, ,#-% �(� /(��,.�#(.1 �

'74 B4@D4=24 (Jn (1))�

n=1 2>=E4A64B C> J (1) � 4<<0 9b)� 0=3 C74A45>A4 8C 8B =>A<�1>D=343�(ix) '74 B4@D4=24 (Sn)

�

n=1 8B =>A<�1>D=343� �=3443� 8C 8<<4380C4;H 5>;;>FB 5A><(iii)� (viii)�(x) '74A4 4G8BCB 0 F40:;H 066;><4A0C8>= ?>8=C S�� >5 (Sn)

�

n=1� �=3443� C78B 8B 0CA8E80; 2>=B4@D4=24 >5 (ix) 0=3 C74 �;0>6;DLB '74>A4<�(xi) '74 066;><4A0C8>= ?>8=C S�� 8B � ��540B81;4 0=3 C74 >1942C8E4 >5 C78B ?A>1;4<

02784E4B C74 >?C8<0; E0;D4 >5 �� 0C S�� =3443� JABC� ;4C DB B44 C70C S�� 8B � ��540B81;4� �>=BCA08=CB >5 � � =>C 8=E>;E8=6 J 0A4 >1E8>DB� '74A4 0A4 C7A44 2>=BCA08=CB8=E>;E8=6 J � 1DC C74H 20= 14 0=0;HI43 F8C7 B8<8;0A <4C7>3B� '7DB� ;4C DB B7>F C70CJ (��1 )� ��4 � ��5 � 0� �8G n0 � �� 5 n0 � n C74=

Jn��n,1

�� 4,n � ��5,n � 0� Jn

��n,1

�� 4,n + ��5,n.

&8=24 Jn� = ��IP� (� |Bn� ) � Jn� F4 70E4 Jn��n,1

��

IP�

��4,n |Bn�

�+ ��

IP�

��5,n |Bn�

��

0=3 C74A45>A4 Jn� (��1 ) � ��IP� (�

��4 |Bn� ) + ��IP� (�

��5 |Bn� )� '7DB� 4<<0 9b) ;403B C>

J (��1 ) = Limn��Jn� (��1 ) �

Limn��IP�(��4 |Bn� ) + Limn��IP�

(��5 |Bn� )

= ��4 + ��5 .

"4GC� ;4C DB B7>F C70C C74 � ��>1942C8E4 4@D0;B 0C S�� =3443� ��IP��Jn��n,1

��=

��IP�

��IP�

�J��n,1

�|Bn

��= ��IP�

�J��n,1

��8<?;84B C70C

n = C � (1 + w)��IP��Jn��n,1

��+ ��n,2 + ��IP�

�R��n,3

�+ ��n,4

+��n,6 + ��IP��R��n,7

�+ ��n,8 + ��n,10 + ��IP�

�R��n,11

�=

C � (1 + w)��IP��J��n,1

��+ ��n,2 + ��IP�

�R��n,3

�+ ��n,4

+��n,6 + ��IP��R��n,7

�+ ��n,8 + ��n,10 + ��IP�

�R��n,11

�.

'74 ;45C 70=3 B834 >5 C74 4@D0;8CH 01>E4 2>=E4A64B C> �B44 (ii)�� B> 8C <DBC 4@D0;4E4AH 066;><4A0C8>= ?>8=C >5 C74 A867C 70=3 B834� �= >C74A F>A3B�

= C � (1 + w)��IP� (J (��1 )) + ��2 + ��IP� (R�

��3 ) + ��4

+��6 + ��IP� (R��7 ) + ��8 + ��10 + ��IP�

�R��11

�.

(xii) �8=0;;H� B8=24 C74 � ��>1942C8E4 0C S�� 4@D0;B C74 ��>1942C8E4 0C D� 1>C7;8=40A >?C8<8I0C8>= ?A>1;4;E01;4 0=3 C74A4 8B => 3D0;8CH 60? 14CF44= C74<��=34AB>= 0=3 "0B7� ��


6. Conclusions

This paper has dealt with the optimal reinsurance problem if both insurer and rein-surer are facing risk and uncertainty, though the classical uncertainty free case is alsoincluded. The levels of uncertainty of insurer and reinsurer do not have to be identi-cal. As a second novelty, the decision variable is not the retained (or ceded) risk, butits sensitivity (mathematical derivative) with respect to the total claims. Thus, if oneimposes strictly positive lower bounds for this variable, the reinsurer moral hazard istotally eliminated. This may be an important property because stop-loss and closelyrelated (often optimal) contracts have been criticized. Indeed, the reinsurer wouldnot accept these solutions in practice due to the lack of incentives of the insurer toverify claims beyond some thresholds.

Three main contributions seem to be reached. Firstly, necessary and su¢ cientoptimality conditions are given, and they apply in all the cases contained in the gen-eral setting above. Secondly, the optimal contract is often a bang-bang solution, i.e.,the sensitivity between the retained risk and the total claims saturates the imposedconstraints. This Önding also explains why stop-loss or related contracts are oftenoptimal if appropriate constraints are not imposed. For some special cases the opti-mal contract might not be a bang-bang one, but there is always a bang-bang contractas close as desired to the optimal one (the optimal reinsurance is in the closure of aset described by bang-bang solutions). Thirdly, the optimal reinsurance problem isequivalent to other linear programming problem (the double-dual problem), despitethe fact that risk and uncertainty (and many pricing principles) cannot be repre-sented by linear expressions. This may be an important Önding for two reasons. Onthe one hand, linear problems are easy to solve in practice, since there are very ef-Öcient algorithms in both Önite and inÖnite dimensions. On the other hand, linearproblems often lead to extreme solutions, which explains why the non linear optimalreinsurance problem may be solved by a bang-bang reinsurance.

Acknowledgments. Research partially supported by Ministerio de Economía,Spain, Grant ECO2012# 39031# C02# 01. The usual caveat applies.

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Optimal reinsurance under risk and uncertainty

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