Diffusion processes and the Fokker-Planck equation (some ideas...)

SDE andFokker-Planck

equation

E. Savin

Introduction

SDE

First-ordersystems

Stochasticintegrals

Diffusionprocesses

Numericalsolutions

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StochasticHamiltoni-ans

,

Diffusion processesand the Fokker-Planck equation (some ideas...)

E. Savin1,2

1Computational Fluid Dynamics Dept.Onera, France

2Mechanical and Civil Engineering Dept.Ecole Centrale Paris, France


equation

E. Savin

Introduction

SDE

First-ordersystems

Stochasticintegrals

Diffusionprocesses

Numericalsolutions

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Outline

1 Introduction

2 Stochastic differential equations (SDE)First-order stochastic systems driven by noiseStochastic integralsDiffusion processes

3 Numerical simulations of SDEStochastic modeling with SDENumerical schemes

4 Stochastic Hamiltonian dynamical systems


equation

E. Savin

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SDE

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Linear or non linear dynamical systems

Discrete linear dynamical system:

M :q `D 9q `Kq “ F ptq ,

or for u “ pq,pq, p “M 9q:

9uptq “

„

0 M´1

´K ´DM´1

uptq `

„

0I

F ptq , up0q “ u0 .

Non-linear dynamical system, e.g. the Duffing oscillator:

M :q `D 9q `Kq `K0q3 “ F ptq ,

or:9uptq “ bpu, tq ` aF ptq , up0q “ u0 ,

with bpu, tq “

ˆ

M´1p´DM´1p´Kq ´K0q

3

˙

, a “

ˆ

01

˙

.


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Example: free vibrations of a single oscillator

9Uptq “d

dt

ˆ

qM 9q

˙

“

„

0 M´1

´K ´DM´1

looooooooomooooooooon

´L

Uptq , Up0q “ U0 ,

where U0 is a r.v. with marginal PDF π0pu0q.

Formally Uptq “ fpU0, tq “ e´LtU0 and the marginalPDF at time t, πpu; tq, is given by the causalityprinciple (lecture #1):

π0pu0q “ πpfpu0, tqqdetp∇ufq .

It yields the conservation equation of the PDF:

0 “dπ0

dt“ Btπ `∇u ¨ pπ 9uq “ Btπ `∇u ¨ Jpπq

with J the probability flux and Jpπq “ ´πLu theconstitutive behavior.


equation

E. Savin

Introduction

SDE

First-ordersystems

Stochasticintegrals

Diffusionprocesses

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Stochasticmodeling

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Outline

1 Introduction





equation

E. Savin

Introduction

SDE

First-ordersystems

Stochasticintegrals

Diffusionprocesses

Numericalsolutions

Stochasticmodeling

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First-order stochastic differential equation

A general first-order stochastic differential equation for theprocess U indexed on R` with values in Rq:

9Uptq “ bpU , tq ` apU , tqF ptq , Up0q “ U0 ,

with the data:

u, t ÞÑ bpu, tq : Rq ˆ R` Ñ Rq the drift function;u, t ÞÑ apu, tq : Rq ˆ R` ÑMq,ppRq the scatteringoperator;U0 is an r.v. in Rq with known marginal PDF π0pu0q;F ptq “ pF1ptq, . . . Fpptqq is a second-order Gaussianrandom process indexed on R with values in Rp, alsocentered, stationary, such that F1ptq, . . . Fpptq aremutually independent and mean-square continuous,with:

SF pωq “ S01r´B,BspωqrIps , S0 ą 0 , B ą 0 .


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First-order systems driven by noise

B ă `8: colored noise, hot topic!

Uptq “ U0 `

ż t

0bpUpsq, sqds`

ż t

0apUpsq, sqF psqds .

B Ñ `8: F Ñ 9W the normalized Gaussian whitenoise, and the solution of the first-oder SDE holds as a“stochastic integral”:

Uptq “ U0 `

ż t

0bpUpsq, sqds`

ż t

0apUpsq, sq ˝ dW psq .

Causality: the family of r.v. tUpsq, 0 ď s ď tu isindependent of the family of r.v. tF pτq, τ ą tu ortdW pτq, τ ą tu.


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White noiseDefinition

Definition

The normalized Gaussian white noise Bptq ” 9W ptqwith values in Rp is the Gaussian stochastic processindexed on R, centered, stationary, with the spectraldensity matrix:

SBpωq “1

2πIp .

Since B1ptq, . . . Bpptq are uncorrelated and jointlyGaussian, they are mutually independent.

B is not second order ~Bptq~2 “ş

TrSBpωqdω “ `8.

This definition holds in the sense of generalized stochastic processes ϕ ÞÑBpϕq : DpT q Ñ L2

pΩ,Rpq where DpT q is the set of C8 functions havinga compact support within T Ď R.


equation

E. Savin

Introduction

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White noiseDefinition

White noise.


equation

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Wiener processDefinition

The white noise is the (generalized) derivative of the Wienerprocess, or Brownian motion.

Definition

The (normalized) Wiener process W ptq with values in Rp isthe stochastic process indexed on R`, s.t.:

W1ptq, . . .Wpptq are mutually independent;

W p0q “ 0 almost surely (a.s.);

If 0 ď s ă t ă `8 let ∆W ps, tq “W ptq ´W psq, then:

@m and 0 ă t1 ă t2 ă ¨ ¨ ¨ ă tm ă `8, W p0q,∆W p0, t1q, ∆W pt1, t2q, ... ∆W ptm´1, tmq are mutuallyindependent r.v. (independent increments);∆W ps, tq is a Gaussian, centered, second-order r.v.with C∆W ps, tq “ pt´ sqIp.


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Wiener processCharacterization

Consequently it can be shown that:

W ptq is a second-order Gaussian, centered, mean-squarecontinuous, non stationary stochastic process;

the covariance and conditional PDF for 0 ď t, s ă `8:

CW pt, sq “ Minpt, sqIp ,

πtpv1; t` s|v; tq “ p2πsq´

p2 e´

v1´v2

2s ;

W ptq has a.s. continuous sample paths;

sample paths t ÞÑW pt, θq, θ P Ωθ, are non differentiablea.s.

As a generalized derivative with dW “ pdW1, . . .dWpq:

dW pϕq “ Bp´ 9ϕq , @ϕ P DpRq .


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Wiener processCharacterization

Wiener process in R3.


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Stochastic integralsDefinition

Let Xptq be a stochastic process indexed by R` witha.s. continuous sample paths.

Assume the r.v. tXpsq, 0 ď s ď tu are independent ofthe r.v. t∆W pt, τq, τ ą tu: a non anticipative process,then

ż t

0XpsqdλW psq

“ l. i.p.KÑ`8

Kÿ

k“1

rp1´ λqXptkq ` λXptk`1qs∆W ptk, tk`1q ,

for any partition 0 “ t1 ă t2 ă ¨ ¨ ¨ ă tK`1 “ t of r0, tswith max

1ďkďKptk`1 ´ tkq Ñ

KÑ`80.


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Stochastic integralsApplication to stochastic differential calculus

A simple example–remind ∆W9∆t12 for the real-valued

Wiener process W :

ż t

0W psqdλW psq “

1

2W ptq2 `

ˆ

λ´1

2

˙

t ,

from which one deduces the stochastic differential:

dλpW ptq2q “ 2W ptqdW ptq ` p1´ 2λqdt .

More generally (λ “ 0 is called the Ito formula):

dλpfpW ptqq “ f 1pW ptqqdW ptq `

ˆ

1

2´ λ

˙

f2pW ptqqdt .


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Stochastic integralsStratonovich-Ito

If λ “ 12 the usual differential calculus applies, and the

solution of SDE holds as a Stratonovich integral (1966):

Uptq “ U0 `

ż t

0bpUpsq, sqds`

ż t

0apUpsq, sq ˝ dW psq .

If λ “ 0, its solution holds as an Ito integral (1944):

Uptq “ U0 `

ż t

0bpUpsq, sqds`

ż t

0apUpsq, sqdW psq ,

where:

bpu, tq “ bpu, tq `1

2aT∇ua .

Uptq is a Markov process.


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Markov processesDefinition

Definition

The conditional probability given t0 ă ¨ ¨ ¨ ă tm ă t:

πtpu; t|u0, . . .um; t0, . . . tmq “πpu0, . . .um,u; t0, . . . tm, tq

πpu0, . . .um; t0, . . . tmq.

Definition

Let Uptq be a stochastic process defined on pΩ, E , P q andindexed on R` with values in Rq. It is a Markov process if:

for all 0 ď t1 ă ¨ ¨ ¨ ă tm ă t and u1, . . .um,u in Rq

πtpu; t|u0, . . .um; t0, . . . tmq “ πtpu; t|um; tmq ;

the marginal PDF π0pu0q of Up0q can be any PDF.


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Markov processesChapman-Kolmogorov equation

A Markov process is fully characterized by:

its marginal PDF πpu; tq,and its transition PDF πtpu; t|v; sq, 0 ď s ă t ă `8,with

πpu; tq “

ż

Rq

πtpu; t|v; sqπpv; sqdv .

πt satisfies the Chapman-Kolmogorov equation:

πtpu; t|u1; t1q “

ż

Rqπtpu; t|v; sqπtpv; s|u1; t1qdv , t1 ă s ă t .

Homogeneous Markov process:

πtpu; t|v; sq “ πtpu; t´ s|v; 0q , 0 ď s ă t ă `8 .

The Brownian motion is a Markov process.


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Diffusion processesDefinition

Definition

The Rq–valued Markov process Uptq with a.s. continuoussample paths and transition PDF πtpv; s|u; tq is a diffusionprocess if @ε ą 0 (but not necessarily small), @u P Rq thefirst moments of its increments are such that for h ą 0:

ż

vúěεπtpdv; t` h|u; tq “ ophq ,

ż

vúăεpv ´ uqπtpdv; t` h|u; tq “ hbpu, tq ` ophq ,

ż

vúăεpv ´ uq b pv ´ uqπtpdv; t` h|u; tq “ hσpu, tq ` ophq ,

where b P Rq and σ PMq,qpRq symmetric, positive.


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Diffusion processesInterpretation

Continuity: particles moving on sample paths of adiffusion process only make small jumps, or theprobability of moving a distance ε goes to zero as h goesto zero no matter how small ε is.

Drift: those particles can have a net mean velocity b.

Diffusion: particles spread as time increases with therate Trσ. Entropy increases while the phase spacecontracts, thus some information (energy) gets lost.

Upt` hq ´Uptq « hbpUptq, tq ` σ12 pUptq, tq∆W pt, t` hq .


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Fokker-Planck equation

The marginal PDF π and transition PDF πt of a diffusionprocess satisfy the Fokker-Planck equation:

Btπ `∇u ¨

ˆ

πb´1

2∇u ¨ pπσq

˙

“ 0 ,

with πpu0; 0q “ π0pu0q and limhÓ0 πtpu; t`h|v; tq “ δpu´vq.ż

RqfpuqBtπtpu; t|v; sqdu “ lim

hÓ0

1

h

ż

Rqfpuq pπtpu; t` h|v; sq ´ πtpu; t|v; sqqdu

“ limhÓ0

1

h

ż

Rqπtpu; t|v; sq

„ż

Rqfpu1qπtpu

1; t` h|u; tqdu1 ´ fpuq

du (C-K)

“ limhÓ0

1

h

ż

Rqπtpu; t|v; sq

ż

Rq

`

fpu1q ´ fpuq˘

πtpu1; t` h|u; tqdu1du (norm.)

“ limhÓ0

1

h

ż

Rqπtpu; t|v; sq

ż

u1´uăε

`

fpu1q ´ fpuq˘

πtpu1; t` h|u; tqdu1du , @f P C 2

0 .

Then use a Taylor expansion for f , definitions of drift and diffusion, andintegrate by parts.


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Ito’s stochastic differential equations (ISDE)Solutions

dU “ bpU , tqdt` apU , tqdW , Up0q “ U0 ,

with the regularity assumptions:

bpu, tq ` apu, tq ď Kp1` uq ,

bpu1, tq ´ bpu, tq ` apu1, tq ´ apu, tq ď Ku1 ´ u .

1 Then the SDE has a unique solution, with a.s.continuous sample paths. If in addition b and a areindependent of t, Uptq is homogeneous.

2 If t ÞÑ bpu, tq and t ÞÑ apu, tq are continuous, Uptq isalso a diffusion process with σ “ aaT.


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Ito’s stochastic differential equations (ISDE)Example: Black-Scholes1 model

The relative variation of a stock Uptq with constant(annualized) drift rate µ and volatility σ:

dU

U“ µdt` σdW , Up0q “ U0 .

Transformation to a Stratonovich SDE:

dU

U“

ˆ

µ´σ2

2

˙

dt` σ ˝ dW , Up0q “ U0 ,

for which “normal rules of integration” apply:

Uptq “ U0 eσW ptq`pµ´σ2

2qt .

The Fokker-Planck equation:

Btπ ` µBupπuq ´σ2

2B2upπu

2q “ 0 , πpu; 0q “ π0puq .

1Fischer Black (1938–1995), Myron Scholes (1941–): American financial economists.

M. Scholes received the Sveriges Riksbank Prize in Economic Sciences in Memory of A.Nobel in 1997 for this model for valuing options, together with Robert Morton (1944–).


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E. Savin

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Outline

1 Introduction





equation

E. Savin

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SDE

First-ordersystems

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First-order stochastic differential equation

A general first-order stochastic differential equation for theprocess U indexed on R` with values in Rq:

#

9Uptq “ bpU , tq ` apU , tqF ptq , t ą 0 ,

Up0q “ U0 ,

with the data:

u, t ÞÑ bpu, tq : Rq ˆ R` Ñ Rq the drift function;

u, t ÞÑ apu, tq : Rq ˆ R` ÑMq,ppRq the scatteringoperator;

U0 is an r.v. in Rq with known marginal PDF π0pu0q;

F ptq “ pF1ptq, . . . Fpptqq is a second-order Gaussianrandom process indexed on R` with values in Rp,centered, mean-square continuous.


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Markovian realizationDefinition

Definition

F ptq indexed on R` with values in Rp, second-order,Gaussian, centered and mean-square continuous admits aMarkovian realization if:

$

&

%

F ptq “HV ptq , t ě 0 ,9V ptq “ PV ptq `QBptq , t ą 0 ,V p0q “ V 0 a.s.

where V 0 is a Gaussian r.v. in Rn, V ptq is a diffusionprocess indexed on R` with values in Rn, P ,Q PMnpRq,H PMp,npRq, <etλjpP qu ă 0.

This is equivalent to a linear Ito stochastic differential equation.

V 0 „ N p0,Σ0q where Σ0 “ş`8

0eτP QQT eτP

T

dτ .


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Physically realizable processDefinition

Definition

F ptq indexed on R with values in Rp, second-order,mean-square stationary and continuous, centered, isphysically realizable if DH P L2pRq, suppH Ď R`, s.t.:

F ptq “

ż t

´8

Hpt´ τqBpτqdτ ,

or equivalently SF pωq “1

2πpHpωqpHpωq˚, @ω P R.

A necessary and sufficient condition (Rozanov 1967):

ż

R

lnpdetSF pωqq

1` ω2dω ą ´8 .


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Markovian realizationExistence for a physically realizable process

Theorem

A necessary and sufficient condition:

SF pωq “RpiωqRpiωq˚

2π|P piωq|2, or Hpωq “

Rpiωq

P piωq,

where:

P pzq is a polynomial of degree d on C with realcoefficients and roots in the half-plane <epzq ă 0,

Rpzq is a polynomial on C with coefficients in Mp,npRqand degree r ă n.

The Markovian realization always exists in infinitedimension n “ `8.


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First-order SDE (cont’d)

A non linear first-order stochastic differential equation forthe process Zptq “ pUptq,V ptqq indexed on R` with valuesin Rν , ν “ q ` n:

#

dZptq “ bzpZ, tqdt` azdW , t ą 0 ,

Zp0q “ Z0 ,

where Z0 “ pU0,V 0q,

bzpu,v, tq “

„

bpu, tq ` apu, tqHvPv

, az “

„

0 00 Q

,

and W ptq is the Wiener process in Rν .


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Numerical integration of SDEStrong convergence

"

dUptq “ bpU, tqdt` apU, tqdW ptq , t ą 0 ,Up0q “ U0 a.s.

Definition

An approximation pUjqj converges with strong order k ą 0 ifDKj ą 0:

E!ˇ

ˇ

ˇUpj∆tq ´ Uj

ˇ

ˇ

ˇ

)

ď Kjp∆tqk .

The sample paths of the approximation U should be close tothose of the Ito process.


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Numerical integration of SDEWeak convergence

"

dUptq “ bpU, tqdt` apU, tqdW ptq , t ą 0 ,Up0q “ U0 a.s.

Definition

An approximation pUjqj converges with weak order k ą 0 iffor any polynomial g DKg,j ą 0:

ˇ

ˇ

ˇE tgpUpj∆tqqu ´ EtgpUjqu

ˇ

ˇ

ˇď Kg,jp∆tq

k .

The probability distribution of the approximation should beclose to that of the Ito process in order to get a goodestimate of the expectation (gpuq “ u) or the variance(gpuq “ u2), for example.


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Time discrete approximationsExplicit 0.5–order methods

Assume that v and a are independent of time t (thusUptq is a diffusion process), and let tj “ j∆t,bj “ bpUjq, aj “ apUjq, U0 „ π0pdu0q, G „ N p0, 1q.Ito SDE: the Euler-Maruyama scheme (1955),

Uj`1 “ Uj ` bj∆t` aj?

∆tG ,

U0 “ U0 .

Stratonovich SDE: the Euler-Heun scheme (1982),

Uj`1 “ Uj ` bj∆t` aj?

∆tG ,

aj “1

2

”

aj ` a´

Uj ` aj?

∆tG¯ı

,

U0 “ U0 .

Both have a strong order k “ 12 (vs. k “ 1 for ordinary

differential equations) and a weak order k “ 1.


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Time discrete approximationsExplicit 1–order methods

The Milstein scheme (1974):

Uj`1 “ Uj ` bλ,j∆t` aj?

∆tG`1

2aja

1j∆tpG

2 ` 2λ´ 1q ,

U0 “ U0 ,

where λ “ 0 (Ito SDE) or λ “ 12 (Stratonovich SDE).

The Runge-Kutta Milstein scheme (1984):

Uj`1 “ Uj ` bλ,j∆t` aj?

∆tG`1

2aj a

1j∆tpG

2 ` 2λ´ 1q ,

aj a1j “ p∆tq

´ 12

”

a´

Uj ` aj?

∆t¯

´ aj

ı

,

U0 “ U0 .

Both have strong and weak orders k “ 1 (under mildconditions on b and a).


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Time discrete approximationsStochastic Taylor approximations

Higher-order schemes may be derived using stochasticTaylor expansions:

Uj`1 ´ Uj “

ż tj`1

tj

bpUqdt`

ż tj`1

tj

apUqdW

»

ż tj`1

tj

`

bpUjq ` b1pUjq∆Uj

˘

dt`

ż tj`1

tj

`

apUjq ` a1pUjq∆Uj

˘

dW ,

where ∆Uj “şt

tjbpUqdτ `

şt

tjapUqdW .

Thenştj`1

tj

şttj

dλW psqdλW ptq “12p∆W q

2 ` pλ´ 12q∆t.

Higher-order expansions involve additional r.v.∆Zj “

ştj`1

tj

şttj

dWdt with Etp∆Zjq2u9∆t3 etc.

Weak Taylor approximations U0 „ U0, ∆W „ ∆W ,∆Zj „ ∆Zj with approximately the same momentproperties.


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Outline

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Canonical equations

Q P Rq the position, P P Rq the momentum, H theHamiltonian (independent of time), F the nonconservative forces,

9Q “∇pHpQ,P q ,9P “ ´∇qHpQ,P q ` F pQ,P , 9W q ,

where F pq,p,fq “ ´fpHqG∇pH` gpHqSf and 9W awhite noise.

Example: Duffing oscillator driven by white noise,

M :Q`D 9Q`KQ`K0Q3 “ g0S0

9W

then Ec “ 12M

9Q2, Ep “ 12KQ

2 ` 14K0Q

4 and P “ B 9qEc,thus:

HpQ,P q “ 1

2M´1P 2 `

1

2KQ2 `

1

4K0Q

4 .


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SDE

First-ordersystems

Stochasticintegrals

Diffusionprocesses

Numericalsolutions

Stochasticmodeling

Numericalschemes


Fokker-Planck equation

The associated Fokker-Planck equation for thetransition PDF πtpq

1,p1; t1|q,p; tq reads:

Btπ ` tπ,Hu ´∇p ¨ Jpπq “ 0 ,

with the Poisson bracket and probability flux beingdefined as:

tπ,Hu “∇qπ ¨∇pH´∇pπ ¨∇qH ,

Jpπq “ π

„

fpHqG`1

2gpHqg1pHqSST

∇pH

`1

2gpHq2SST∇pπ .


equation

E. Savin

Introduction

SDE

First-ordersystems

Stochasticintegrals

Diffusionprocesses

Numericalsolutions

Stochasticmodeling

Numericalschemes


Summary

What’s new?

Non linear filtering of white noise,A (high-dimensional) PDE for the marginal andtransition PDF of diffusion processes,Stochastic integrals,Numerical simulations of SDE,Application to non linear dynamical systems.

What’s left?

Numerical solutions of the FKE,Computation of second-order quantities of diffusionprocesses.


equation

E. Savin

Introduction

SDE

First-ordersystems

Stochasticintegrals

Diffusionprocesses

Numericalsolutions

Stochasticmodeling

Numericalschemes


Further reading...

A. Friedman: Stochastic Differential Equations and Applications,vol. I & II, Academic Press (1975);

P.E. Kloeden, E. Platen: Numerical Solution of StochasticDifferential Equations, 3rd ed., Springer (1999);

B.K. Øksendal: Stochastic Differential Equations: An Introductionwith Applications, 6th ed., Springer (2003);

C. Soize: The Fokker-Planck Equation for Stochastic DynamicalSystems and its Explicit Steady State Solutions, World Scientific(1994);

D. Talay: Simulation of stochastic differential systems. InProbabilistic Methods in Applied Physics (P. Kree & W. Wedig,eds.), pp. 54-96. Lecture Notes in Physics 451, Springer (1995).

http://store.doverpublications.com/0486453596.html

http://store.doverpublications.com/0486453596.html

http://dx.doi.org/10.1007/978-3-662-12616-5

http://dx.doi.org/10.1007/978-3-662-12616-5

http://dx.doi.org/10.1007/978-3-642-14394-6

http://dx.doi.org/10.1007/978-3-642-14394-6

http://dx.doi.org/10.1142/9789814354110

http://dx.doi.org/10.1142/9789814354110

http://dx.doi.org/10.1142/9789814354110

http://dx.doi.org/10.1007/3-540-60214-3_51

http://dx.doi.org/10.1007/3-540-60214-3_51

http://dx.doi.org/10.1007/3-540-60214-3_51

Diffusion processes and the Fokker-Planck equation (some ideas...)

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