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Stochastic differential equations Solution as a Markov process

cours ARO07–MSSD #4 Random Models of Dynamical Systems Introduction to SDE’s Stochastic differential equations

Fran¸ cois Le Gland INRIA Rennes + IRMAR people.rennes.inria.fr/Francois.Le_Gland/insa-rennes/ 8 December 2020, via Zoom

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Stochastic differential equations Solution as a Markov process

Stochastic differential equations introduction, existence and uniqueness additional properties extension by localization Solution as a Markov process

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Stochastic differential equations Solution as a Markov process

Definition, assumptions on the coefficients

consider the equation Xptq “ Xp0q ` ż t bps, Xpsqq ds ` ż t σps, Xpsqq dBpsq with a m–dimensional Brownian motion B “ pBptq , t ě 0q, and time–dependent coefficients: ‚ a d–dimensional drift vector bpt, xq defined on r0, 8q ˆ Rd ‚ a d ˆ m diffusion matrix σpt, xq defined on r0, 8q ˆ Rd global Lipschitz condition: there exists a positive constant L ą 0 such that for any t ě 0 and any x, x1 P Rd |bpt, xq´bpt, x1q| ď L |x ´x1| and }σpt, xq´σpt, x1q} ď L |x ´x1| linear growth condition: there exists a positive constant K ą 0 such that for any t ě 0 and any x P Rd |bpt, xq| ď K p1 ` |x|q and }σpt, xq} ď K p1 ` |x|q

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Stochastic differential equations Solution as a Markov process

a solution to the SDE is any process X in M2pr0, Tsq such that the identity holds almost surely the condition that X is in M2pr0, Tsq makes sure that the stochastic integral ż t σps, Xpsqq dBpsq defines a (true, square–integrable) martingale: indeed, the vector–valued stochastic integral makes sense iff for any v P Rd, the one–dimensional stochastic integral v ˚ ż t σps, Xpsqq dBpsq “ ż t v ˚ σps, Xpsqq dBpsq makes sense, i.e. iff E ż T }σps, Xpsqq σ˚ps, Xpsqq} ds ă 8 and note that E ż T }σps, Xpsqq σ˚ps, Xpsqq} ds ď 2 K 2 E ż T p1 ` |Xpsq|2q ds

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Stochastic differential equations Solution as a Markov process

Lemma [Gronwall lemma] if the nonnegative function uptq satisfies the functional relation: for any t ě 0 and for some nonnegative constants a, c ě 0 uptq ď a ` c ż t upsq ds then for any t ě 0 uptq ď a exptc tu Proof assume c ą 0 without loss of generality, and note that d dt rexpt´c tu ż t upsq dss “ expt´c tu ruptq´c ż t upsq dss ď a expt´c tu integration yields expt´c tu ż t upsq ds ď a ż t expt´c su ds “ a c p1 ´ expt´c tuq hence ż t upsq ds ď a c pexptc tu ´ 1q l

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Stochastic differential equations Solution as a Markov process

Lemma [sequential Gronwall lemma] if the nonnegative functions unptq satisfies the functional relation: for any t ě 0 and any n ě 1 and for some nonnegative constants a, c ě 0 unptq ď a ` c ż t un´1psq ds with u0ptq ” ¯ u then for any t ě 0 and any n ě 1 unptq ď a exptc tu ` pc tqn n! ¯ u Remark the following uniform estimate holds max

ně1 unptq ď a exptc tu ` rmax ně1

pc tqn n! s ¯ u and asymptotically lim sup

nÑ8 unptq ď a exptc tu

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Stochastic differential equations Solution as a Markov process

Proof actually, the following stronger estimate is proved by induction unptq ď a r1 ` c t ` ¨ ¨ ¨ ` pc tqn´1 pn ´ 1q!s ` pc tqn n! ¯ u ď a exptc tu ` pc tqn n! ¯ u clearly, the estimate holds for n “ 1 assuming that the estimate holds at stage pn ´ 1q, then unptq “ a ` c ż t un´1psq ds ď a r1 ` c ż t r1 ` c s ` ¨ ¨ ¨ ` pc sqn´2 pn ´ 2q!s ds s ` rc ż t pc sqn´1 pn ´ 1q! ds s ¯ u “ a r1 ` c t ` ¨ ¨ ¨ ` pc tqn´1 pn ´ 1q!s ` pc tqn n! ¯ u i.e. the estimate holds at stage n l

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Stochastic differential equations Solution as a Markov process

simple (yet useful) formula | ż t ψpsq ds|p ď tp´1 ż t |ψpsq|p ds hence (taking ψpsq “ φ2psq and using p{2 in place of p) p ż t |φpsq|2 dsqp{2 ď tp{2´1 ż t |φpsq|p ds Proof using the H¨

• lder inequality for conjugate exponents p, p1 yields

| ż t ψpsq ds| ď p ż t 1p1 dsq1{p1 p ż t |ψpsq|p dsq1{p and note that p{p1 “ p ´ 1 l

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Stochastic differential equations Solution as a Markov process

Existence and uniqueness of a solution

Theorem 1 under the global Lipschitz and linear growth conditions, and for any square–integrable initial condition Xp0q, there exists a unique solution to the SDE Xptq “ Xp0q ` ż t bps, Xpsqq ds ` ż t σps, Xpsqq dBpsq Proof uniqueness: let X “ pXptq , t ě 0q and X 1 “ pX 1ptq , t ě 0q be two solutions, with the same initial condition Xp0q “ X 1p0q by difference, for any 0 ď t ď T |Xptq ´ X 1ptq| ď | ż t bps, Xpsqq ´ bps, X 1psqq ds | ` | ż t pσps, Xpsqq ´ σps, X 1psqqq dBpsq |

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Stochastic differential equations Solution as a Markov process

hence E|Xptq ´ X 1ptq|2 ď 2 E| ż t pbps, Xpsqq ´ bps, X 1psqqq ds |2 ` 2 E| ż t pσps, Xpsqq ´ σps, X 1psqqq dBpsq |2 ď 2 t E ż t |bps, Xpsqq ´ bps, X 1psqq|2 ds ` 2 E ż t }σps, Xpsqq ´ σps, X 1psqq}2 ds ď 2 L2 pT ` 1q ż t E|Xpsq ´ X 1psq|2 ds it follows from the Gronwall lemma that for any 0 ď t ď T E|Xptq ´ X 1ptq|2 “ 0

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Stochastic differential equations Solution as a Markov process

Picard iteration: for n “ 0, let X0ptq ” Xp0q for any 0 ď t ď T, and for any n ě 1 consider the Itˆ

• process

Xnptq “ Xp0q ` ż t bps, Xn´1psqq ds ` ż t σps, Xn´1psqq dBpsq no localization is needed here, thanks to the following a priori estimate: there exists a positive constant MpTq such that for any n ě 1 sup

0ďtďT

E|Xnptq|2 ď MpTq (‹) clearly, the estimate holds for n “ 0, and by induction if the estimate holds at stage pn ´ 1q, then E ż t }σps, Xn´1psqq σ˚ps, Xn´1psqq} ds ď K 2 E ż t p1 ` |Xn´1psq|q2 ds ď 2 K 2 pt ` E ż t |Xn´1psq|2 dsq in other words: the integrand s ÞÑ σps, Xn´1psqq belongs to M2pr0, Tsq

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Stochastic differential equations Solution as a Markov process

a priori estimate: |Xnptq| ď |Xp0q| ` | ż t bps, Xn´1psqq ds| ` | ż t σps, Xn´1psqq dBpsq| and E|Xnptq|2 ´ 3 E|Xp0q|2 ď 3 E| ż t bps, Xn´1psqq ds|2 ` 3 E| ż t σps, Xn´1psqq dBpsq|2 ď 3 t E ż t |bps, Xn´1psqq|2 ds ` 3 E ż t }σps, Xn´1psqq}2 ds ď 6 K 2 t E ż t p1 ` |Xn´1psq|2q ds ` 6 K 2 E ż t p1 ` |Xn´1psq|2q ds ď 6 K 2 T pT ` 1q ` 6 K 2 pT ` 1q ż t E|Xn´1psq|2 ds

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Stochastic differential equations Solution as a Markov process

in other words, the sequence unptq “ E|Xnptq|2 satisfies the functional relation unptq ď apTq ` cpTq ż t un´1psq ds with u0ptq ” E|Xp0q|2 it follows from the sequential Gronwall lemma that sup

0ďtďT

E|Xnptq|2 ď apTq exptcpTq Tu ` pcpTq Tqn n ! E|Xp0q|2 which proves the a priori estimate (‹) where the bound MpTq “ apTq exptcpTq Tu ` max

ně1 rpcpTq Tqn

n ! s E|Xp0q|2 depends on T, K and E|Xp0q|2, and does not depend on L

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Stochastic differential equations Solution as a Markov process

uniform a priori estimate: |Xnpsq| ď |Xp0q| ` | ż s bpu, Xn´1puqq du| ` | ż s σpu, Xn´1puqq dBpuq| uniform upper bound sup

0ďsďt

|Xnpsq| ď |Xp0q| ` sup

0ďsďt

| ż s bpu, Xn´1puqq du| ` sup

0ďsďt

| ż s σpu, Xn´1puqq dBpuq|

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Stochastic differential equations Solution as a Markov process

using the Doob inequality yields Er sup

0ďsďt

|Xnpsq|2 s ´ 3 E|Xp0q|2 ď 3 Er sup

0ďsďt

| ż s bpu, Xn´1puqq du|2 s ` 3 Er sup

0ďsďt

| ż s σpu, Xn´1puqq dBpuq|2 s ď 3 t Er sup

0ďsďt

ż s |bpu, Xn´1puqq|2 du s ` 12 E ż t }σps, Xn´1psqq}2 ds ď 6 K 2 t E ż t p1 ` |Xn´1psq|2q ds ` 24 K 2 E ż t p1 ` |Xn´1psq|2q ds ď 6 K 2 T pT ` 4q ` 6 K 2 pT ` 4q ż t Er sup

0ďuďs

|Xn´1puq|2 s ds

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Stochastic differential equations Solution as a Markov process

in other words, the sequence ¯ unptq “ Er sup

0ďsďt

|Xnpsq|2 s satisfies the functional relation ¯ unptq ď ¯ apTq ` ¯ cpTq ż t ¯ un´1psq ds with ¯ u0ptq ” E|Xp0q|2 it follows from the sequential Gronwall lemma that the stronger uniform a priori estimate holds Er sup

0ďtďT

|Xnptq|2 s ď ¯ apTq expt¯ cpTq Tu ` max

ně1 rp¯

cpTq Tqn n ! s E|Xp0q|2 where the bound depends on T, K and E|Xp0q|2, and does not depend on L

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Stochastic differential equations Solution as a Markov process

existence: back to the Picard iteration, by difference Xn`1psq ´ Xnpsq “ ż s pbpu, Xnpuqq ´ bpu, Xn´1puqqq du ` ż s pσpu, Xnpuqq ´ σpu, Xn´1puqqq dBpuq hence sup

0ďsďt

|Xn`1psq ´ Xnpsq| ď sup

0ďsďt

| ż s pbpu, Xnpuqq ´ bpu, Xn´1puqqq du| ` sup

0ďsďt

| ż s pσpu, Xnpuqq ´ σpu, Xn´1puqqq dBpuq| introduce the function εnptq “ Er sup

0ďsďt

|Xnpsq ´ Xn´1psq|2s

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Stochastic differential equations Solution as a Markov process

using the Doob inequality yields εn`1ptq ď 2 Er sup

0ďsďt

| ż s pbpu, Xnpuqq ´ bpu, Xn´1puqqq du|2s ` 2 Er sup

0ďsďt

| ż s pσpu, Xnpuqq ´ σpu, Xn´1puqqq dBpuq|2s ď 2 t Er sup

0ďsďt

ż s |bpu, Xnpuqq ´ bpu, Xn´1puqq|2 dus ` 8 Er ż t }σpu, Xnpuqq ´ σpu, Xn´1puqq}2 dus ď 2 L2 pT ` 4q Er ż t |Xnpsq ´ Xn´1psq|2 dss ď 2 L2 pT ` 4q ż t Er sup

0ďuďs

|Xnpuq ´ Xn´1puq|2s ds

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Stochastic differential equations Solution as a Markov process

in other words, the sequence εnptq satisfies the functional relation εn`1ptq ď cpTq ż t εnpsq ds by induction, for any 0 ď t ď T Er sup

0ďtďT

|Xn`1ptq ´ Xnptq|2s ď pcpTq Tqn n ! ε1pTq using the Markov inequality yields Pr sup

0ďtďT

|Xn`1ptq ´ Xnptq| ą 2´pn`1qs ď 4n`1 Er sup

0ďtďT

|Xn`1ptq ´ Xnptq|2s ď p4 cpTq Tqn n ! 4 ε1pTq it follows from the Borel–Cantelli lemma that, almost surely sup

0ďtďT

|Xn`1ptq ´ Xnptq| ď 2´pn`1q for any n ě n˚

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Stochastic differential equations Solution as a Markov process

the triangle inequality yields sup

0ďtďT

|Xn`pptq ´ Xnptq| ď

p

ÿ

k“1

r sup

0ďtďT

|Xn`kptq ´ Xn`k´1ptq|s ď 2´n almost surely, the sequence Xn is a Cauchy sequence in Cpr0, Tsq, hence the continuous mapping t ÞÑ Xnptq converges uniformly on r0, Ts to a continuous mapping t ÞÑ χptq clearly Xnptq Ñ χptq and ż t bps, Xn´1psqq ds Ñ ż t bps, χpsqq ds in L2 as n Ò 8, and the limit χ satisfies the estimate (‹), so that the integrand s ÞÑ σps, χpsqq belongs to M2pr0, Tsq, hence ż t σps, Xn´1psqq dBpsq Ñ ż t σps, χpsqq dBpsq in L2 as n Ò 8: i.e. the limiting mapping t ÞÑ χptq solves the SDE l

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Stochastic differential equations Solution as a Markov process

Higher moments of the solution

Theorem 2 * if the initial condition Xp0q has a finite p–th order moment for some p ě 2, then the solution Xptq has a finite p–th order moment, for any 0 ď t ď T Hint For any n ě 1, define the stopping time τn “ inft0 ď t ď T : |Xptq| ě nu if such time exists, and τn “ T otherwise then use the Burkholder–Davis–Gundy inequalities to bound the p–th

• rder moment of the stopped process Xnptq “ Xpt ^ τnq, the Gronwall

lemma, and finally let n Ò 8

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Stochastic differential equations Solution as a Markov process

Continuity w.r.t. initial condition

Theorem 3 almost surely, the solution of the SDE depends continuously

• n the initial condition

Proof let X xptq “ x ` ż t bps, X xpsqq ds ` ż t σps, X xpsqq dBpsq and X yptq “ y ` ż t bps, X ypsqq ds ` ż t σps, X ypsqq dBpsq be two solutions of the same equation, with the same Brownian sample path and starting at time t “ 0 from two different initial conditions x ‰ y by difference X xptq ´ X yptq “ x ´ y ` ż t pbps, X xpsqq ´ bps, X ypsqq ds ` ż t pσps, X xpsqq ´ σps, X ypsqqq dBpsq

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Stochastic differential equations Solution as a Markov process

using the Burkholder–Davis–Gundy inequality for some p ě 2 yields E|X xptq ´ X yptq|p ´ 3p´1 |x ´ y|p ď 3p´1 E| ż t pbps, X xpsqq ´ bps, X ypsqqq ds |p ` 3p´1 E| ż t pσps, X xpsqq ´ σps, X ypsqqq dBpsq |p ď 3p´1 tp´1 Er ż t |bps, X xpsqq ´ bps, X ypsqq|p dss ` 3p´1 tp{2´1 Cp Er ż t }σps, X xpsqq ´ σps, X ypsqq}p dss ď 3p´1 T p{2´1 pT p{2 ` Cpq Lp Er ż t |X xpsq ´ X ypsq|p dss it follows from the Gronwall lemma that E|X xptq ´ X yptq|p ď 3p´1 |x ´ y|p exptcpTq tu

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Stochastic differential equations Solution as a Markov process

this estimate holds for any p ě 2, in particular it holds for p “ d ` α, and it follows from (a generalization of) the Kolmogorov continuity criterion that the mapping x ÞÑ X xptq is continuous almost surely (here t ě 0 is fixed, but one could prove that the mapping pt, xq ÞÑ X xptq is jointly continuous almost surely) l Remark [Feller continuity] for any bounded continuous function f defined

• n Rd, the mapping x ÞÑ Erf pX xptqqs is continuous (using the Lebesgue

dominated convergence theorem) Remark * for any t ě 0 and any x P Rd the solution X xptq depends measurably on pBpsq , 0 ď s ď tq

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Stochastic differential equations Solution as a Markov process

Stochastic differential equations and martingales

Proposition 4 let X be the unique solution of the SDE Xptq “ Xp0q ` ż t bpXpsqq ds ` ż t σpXpsqq dBpsq (with time–independent coefficients), and let f be a twice differentiable function, with bounded first derivative, then Mf ptq “ f pXptqq ´ f pXp0qq ´ ż t L f pXpsqq ds is a martingale: here L f pxq “ f 1pxq bpxq ` 1

2 tracepf 2pxq apxqq

with apxq “ σ σ˚pxq i.e. L is the second–order partial differential operator L “

d

ÿ

i“1

bip¨q B Bxi ` 1

2 d

ÿ

i,j“1

ai,jp¨q B2 Bxi Bxj

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Stochastic differential equations Solution as a Markov process

Proof writing the Itˆ

• formula for the Itˆ
• process

Xptq “ Xp0q ` ż t bpXpsqq ds ` ż t σpXpsqq dBpsq and for the function f , yields f pXptqq “ f pXp0qq ` ż t f 1pXpsqq σpXpsqq dBpsq ` ż t pf 1pXpsqq bpXpsqq ` 1

2 tracerf 2pXpsqq apXpsqqsq ds

hence f pXptqq ´ f pXp0qq ´ ż t L f pXpsqq ds “ ż t f 1pXpsqq σpXpsqq dBpsq is a martingale, since the integrand s ÞÑ f 1pXpsqq σpXpsqq is in M2pr0, Tsq l

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Stochastic differential equations Solution as a Markov process

Kramers approximation

motion of a particle of mass m in a force field bpxq ` σ 9 w and subject to a friction proportional to velocity, defined by the Newton law as m : xpsq “ ´α 9 xpsq ` bpxpsqq ` σ 9 wpsq with initial conditions xp0q “ q and 9 xp0q “ p second order differential equation: rewriting more rigorously as a system

• f first order SDE’s yields

Xptq “ q ` ż t V psq ds V ptq “ p ` 1 m r´α ż t V psq ds ` ż t bpXpsqq ds ` σ Bptqs where the deterministic force field satisfy the usual conditions for existence and uniqueness, i.e. global Lipschitz continuity and linear growth at infinity

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Stochastic differential equations Solution as a Markov process

consider the SDE obtained in the limit as m goes to zero, i.e. X˚ptq “ q ` 1 α ż t bpX˚psqq ds ` σ α Bptq Theorem 5 the process Xptq provides a smooth (differentiable) approximation of the process X˚ptq uniformly on r0, Ts, i.e. sup

0ďtďT

|Xptq ´ X˚ptq| Ñ 0 almost surely as m goes to zero Remark apparently, something could go wrong with V ptq when m goes to zero, because of the 1 m factor actually, everything is under control, thanks to the negative factor ´α

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Stochastic differential equations Solution as a Markov process

Illustration

deterministic force field bpxq “ ´λ x yields Xptq “ q ` ż t V psq ds V ptq “ p ` 1 m r´α ż t V psq ds ´ λ ż t Xpsq ds ` σ Bptqs a linear Gaussian SDE ˜ Xptq V ptq ¸ “ ˜ q p ¸ ` 1 m r ż t ˜ 0 I ´λ I ´α I ¸ ˜ Xpsq V psq ¸ ds ` ˜ 0 σ I ¸ Bptq s and the SDE obtained in the limit as m goes to zero, is X˚ptq “ q ´ λ α ż t X˚psq ds ` σ α Bptq

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Stochastic differential equations Solution as a Markov process

Local uniqueness

for i “ 1, 2, under the global Lipschitz and linear growth conditions, and for any square–integrable initial conditions, there exists a unique solution to the SDE Xiptq “ Xip0q ` ż t bips, Xipsqq ds ` ż t σips, Xipsqq dBpsq under the local agreement condition, i.e. if there exists an open subset D Ă Rm such that ‚ the coefficients coincide on D, i.e. for any t ě 0 and any x P D b1pt, xq “ b2pt, xq and σ1pt, xq “ σ2pt, xq ‚ the initial conditions coincide on D, i.e. if X1p0q P D, then X2p0q P D and X2p0q “ X1p0q (and conversely) then ‚ the two solutions leave D at the same time ‚ and they coincide until the first time where they both leave D

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Stochastic differential equations Solution as a Markov process

Theorem 6 * for i “ 1, 2, introduce the stopping time τi “ inft0 ď t ď T : Xiptq R Du if such time exists, and τi “ T otherwise then, under the local agreement condition τ1 “ τ2 and sup

0ďtďτ1

|X1ptq ´ X2ptq| “ 0 almost surely Proof for i “ 1, 2, introduce the indicator r.v. φiptq “ 1tXipsq P D for any 0 ď s ď tu “ 1tτi ě tu and note that the mapping t ÞÑ φiptq is nonincreasing clearly φ1ptq pX1p0q ´ X2p0qq “ 0

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Stochastic differential equations Solution as a Markov process

by difference φ1ptq pX1ptq ´ X2ptqq “ φ1ptq ż t pb1ps, X1psqq ´ b2ps, X1psqqq ds ` φ1ptq ż t pb2ps, X1psqq ´ b2ps, X2psqqq ds ` φ1ptq ż t pσ1ps, X1psqq ´ σ2ps, X1psqqq dBpsq ` φ1ptq ż t pσ2ps, X1psqq ´ σ2ps, X2psqqq dBpsq note that if t ď τ1 then s ď τ1 for any 0 ď s ď t: firstly ż t pb1ps, X1psqq ´ b2ps, X1psqqq ds “ ż t φ1psq pb1ps, X1psqq ´ b2ps, X1psqqq ds “ 0

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Stochastic differential equations Solution as a Markov process

secondly, if t ď τ1 then ż t pσ1ps, X1psqq ´ σ2ps, X1psqqq dBpsq “ ż t^τ1 pσ1ps, X1psqq ´ σ2ps, X1psqqq dBpsq “ ż t φ1psq pσ1ps, X1psqq ´ σ2ps, X1psqqq dBpsq “ 0 therefore, it remains only φ1ptq pX1ptq ´ X2ptqq “ φ1ptq ż t pb2ps, X1psqq ´ b2ps, X2psqqq ds ` φ1ptq ż t pσ2ps, X1psqq ´ σ2ps, X2psqqq dBpsq

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Stochastic differential equations Solution as a Markov process

and φ1ptq |X1ptq ´ X2ptq|2 ď 2 t φ1ptq ż t |b2ps, X1psqq ´ b2ps, X2psqqq|2 ds ` 2 φ1ptq | ż t pσ2ps, X1psqq ´ σ2ps, X2psqqq dBpsq |2 thirdly, if t ď τ1 then ż t |b2ps, X1psqq ´ b2ps, X2psqqq|2 ds “ ż t φ1psq |b2ps, X1psqq ´ b2ps, X2psqq |2 ds

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Stochastic differential equations Solution as a Markov process

fourthy, if t ď τ1 then ż t pσ2ps, X1psqq ´ σ2ps, X2psqqq dBpsq “ ż t^τ1 pσ2ps, X1psqq ´ σ2ps, X2psqqq dBpsq “ ż t φ1psq pσ2ps, X1psqq ´ σ2ps, X2psqqq dBpsq hence φ1ptq |X1ptq ´ X2ptq|2 ď 2 t ż t φ1psq |b2ps, X1psqq ´ b2ps, X2psqqq|2 ds ` 2 | ż t φ1psq pσ2ps, X1psqq ´ σ2ps, X2psqqq dBpsq |2

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Stochastic differential equations Solution as a Markov process

taking expectation and using the global Lipschitz condition yields Er φ1ptq |X1ptq ´ X2ptq|2 s ď 2 t E ż t φ1psq |b2ps, X1psqq ´ b2ps, X2psqqq|2 ds ` 2 E ż t φ1psq }σ2ps, X1psqq ´ σ2ps, X2psqq}2 ds ď 2 L2 pT ` 1q ż t Er φ1psq |X1psq ´ X2psq|2 s ds it follows from the Gronwall lemma that for any 0 ď t ď T Er φ1ptq |X1ptq ´ X2ptq|2 s “ 0 hence φ1ptq |X1ptq ´ X2ptq| “ 0 almost surely

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Stochastic differential equations Solution as a Markov process

by continuity of sample paths sup

0ďtďT

φ1ptq |X1ptq ´ X2ptq| “ 0 almost surely it follows that X2ptq “ X1ptq almost surely if 0 ď t ď τ1, hence τ2 ě τ1, and similarly τ1 ě τ2 l

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Stochastic differential equations Solution as a Markov process

Existence and uniqueness of a solution, by localization

replace the global Lipschitz condition by local Lipschitz condition: for any integer N ě 1, there exists a positive constant LN ą 0 such that for any t ě 0 and any x, x1 P Bp0, Nq |bpt, xq´bpt, x1q| ď LN |x´x1| and }σpt, xq´σpt, x1q} ď LN |x´x1| Theorem 7 under the local Lipschitz and linear growth conditions, and for any square–integrable initial condition Xp0q, there exists a unique solution to the SDE Xptq “ Xp0q ` ż t bps, Xpsqq ds ` ż t σps, Xpsqq dBpsq

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Stochastic differential equations Solution as a Markov process

Proof for any integer N ě 1, introduce the square–integrable initial condition XNp0q “ # Xp0q if |Xp0q| ď N if |Xp0q| ą N and the coefficients bNpt, xq “ $ ’ ’ & ’ ’ % bpt, xq if |x| ď N bpt, xq p2 ´ |x| N q if N ă |x| ď 2 N if |x| ą 2 N σNpt, xq “ $ ’ ’ & ’ ’ % σpt, xq if |x| ď N σpt, xq p2 ´ |x| N q if N ă |x| ď 2 N if |x| ą 2 N these coefficients satisfy the global Lipschitz and linear growth conditions

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Stochastic differential equations Solution as a Markov process

there exists a unique solution in M2pr0, Tsq to the SDE XNptq “ XNp0q ` ż t bNps, XNpsqq ds ` ż t σNps, XNpsqq dBpsq and the uniform a priori estimate Er sup

0ďtďT

|XNptq|2 s ď MpTq holds, with a constant MpTq that does not depend on N introduce the stopping time τN “ inft0 ď t ď T : |XNptq| ě Nu if such time exists, and τN “ T otherwise, and the good event ΩN “ t sup

0ďtďT

|XNptq| ă Nu Ă tτN “ Tu

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Stochastic differential equations Solution as a Markov process

if N ě M, then it follows from the local uniqueness theorem that XNptq “ XMptq almost surely for any 0 ď t ď τM

• n the good event ΩM the sample path pXNptq , 0 ď t ď Tq coincides

with the sample path pXMptq , 0 ď t ď Tq, hence converges to a limit as N Ò 8 clearly, the sequence ΩM is nondecreasing and using the Markov inequality yields PrΩzΩMs “ Pr sup

0ďtďT

|XMptq| ě Ms ď 1 M2 Er sup

0ďtďT

|XMptq|2 s ď 1 M2 MpTq hence the sequence ΩM converges to Ω therefore, the sequence XNptq converges on the event Ω as N Ò 8, to a limit χptq that coincides with XMptq on each good event ΩM

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Stochastic differential equations Solution as a Markov process

• n the good event ΩN it holds |XNpsq| ď N for any 0 ď s ď t, hence

ż t bNps, XNpsqq ds “ ż t bps, XNpsqq ds “ ż t bps, χpsqq ds and ż t σNps, XNpsqq dBpsq “ ż t^τN σNps, XNpsqq dBpsq “ ż t 1ts ď τNu σNps, XNpsqq dBpsq “ ż t 1ts ď τNu σps, XNpsqq dBpsq “ ż t 1ts ď τNu σps, χpsqq dBpsq “ ż t^τN σps, χpsqq dBpsq “ ż t σps, χpsqq dBpsq

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Stochastic differential equations Solution as a Markov process

moreover XNptq “ χptq and XNp0q “ Xp0q therefore, on each good event ΩN it holds χptq “ Xp0q ` ż t bps, χpsqq ds ` ż t σps, χpsqq dBpsq in other words, the limiting mapping t ÞÑ χptq solves the SDE uniqueness is proved along the same lines as in the proof of the local uniqueness theorem l

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Stochastic differential equations Solution as a Markov process

Stochastic differential equations Solution as a Markov process

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Stochastic differential equations Solution as a Markov process

Definition a stochastic process X “ pXptq , t ě 0q is a Markov process w.r.t. the filtration F “ pFptq , t ě 0q iff for any t, s ě 0 and for any Borel subset A P BpRdq PrXpt ` sq P A | Fptqs “ PrXpt ` sq P A | Xptqs the Markov process is time–homogeneous (or simply homogeneous), if PrXpt ` sq P A | Xptq “ xs “ Qps, x, Aq

• r equivalently

Erf pXpt ` sqq | Xptq “ xs “ pQpsq f qpxq “ ż Qps, x, dx1q f px1q does not depend on t ě 0 (depends only on x P Rd, s ě 0 and A P BpRdq)

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Stochastic differential equations Solution as a Markov process

Proposition 8 the one–parameter family of operators pQpsq , s ě 0q associated with a Markov process satisfies the semigroup (Chapman–Kolmogorov) equation: for any t, s ě 0 Qpt ` sq “ Qptq Qpsq seen as composition of operators Proof for any bounded measurable function f Erf pXpt ` sqq | Fptqs “ pQpsq f qpXptqq hence, taking expectation of both sides conditionally w.r.t. Fp0q Erf pXpt ` sqq | Fp0qs “ ErpQpsq f qpXptqq | Fp0qs “ pQptq pQpsq f qqpXp0qq while by definition Erf pXpt ` sqq | Fp0qs “ pQpt ` sq f qpXp0qq l

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Stochastic differential equations Solution as a Markov process

Example any stochastic process X with independent and stationary increments (e.g. the Brownian motion) is an homogeneous Markov process Proof by definition the r.v. Xpt ` sq ´ Xptq is independent of Fptq, hence PrpXpt ` sq ´ Xptqq P A, Xptq P B | Fptqs “ PrpXpt ` sq ´ Xptqq P As 1tXptq P Bu and for any bounded measurable function φ ErφpXpt ` sqq | Fptqs “ ErφppXpt ` sq ´ Xptqq ` Xptqq | Fptqs “ ż φpz ` Xptqq PrpXpt ` sq ´ Xptqq P dzs “ ż φpz ` Xptqq µps, dzq where the result depends only on Xptq

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Stochastic differential equations Solution as a Markov process

moreover, the relation ErφpXpt ` sqq | Xptq “ xs “ ż φpz ` xq µps, dzq defines the semigroup Qpsq implicitly, i.e. pQpsq φqpxq “ ż φpz ` xq µps, dzq l

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Stochastic differential equations Solution as a Markov process

Markov property of the solution

Theorem 9 the unique solution of the SDE Xptq “ Xp0q ` ż t bpXpsqq ds ` ż t σpXpsqq dBpsq is a Markov process Proof let t ě 0 be fixed, and note that the refreshed process ¯ Bpsq “ Bpt ` sq ´ Bptq is another standard Brownian motion, independent of Fptq, and the SDE Y psq “ Xptq ` ż s bpY puqq du ` ż s σpY puqq d ¯ Bpuq has a unique solution, while the change of variable v “ t ` u yields Xpt ` sq “ Xptq ` ż t`s

t

bpXpvqq dv ` ż t`s

t

σpXpvqq dBpvq “ Xptq ` ż s bpXpt ` uqq du ` ż s σpXpt ` uqq d ¯ Bpuq

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Stochastic differential equations Solution as a Markov process

by uniqueness of the solution Xpt ` sq “ Y psq “ FpXptq, p ¯ Bpuq , 0 ď u ď sqq therefore Erf pXpt ` sqq | Fptqs “ Erf pFpXptq, p ¯ Bpuq , 0 ď u ď sqqq | Fptqs “ Erf pFpx, p ¯ Bpuq , 0 ď u ď sqqqs|x “ Xptq where the result depends only on Xptq the semigroup is defined for any bounded measurable function f as pQpsq f qpxq “ Erf pXpt ` sqq | Xptq “ xs “ Erf pX xpsqqs l

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Stochastic differential equations Solution as a Markov process

Remark if f is a twice differentiable function, with bounded first derivative, recall that f pXptqq ´ f pXp0qq ´ ż t L f pXpsqq ds is a martingale, hence Erf pXptqqs ´ Erf pXp0qqs “ Er ż t L f pXpsqq dss “ ż t ErL f pXpsqqs ds introducing the probability distribution µpt, dxq “ PrXptq P dxs of the random variable Xptq, defined by xµptq, φy “ ż

E

µpt, dxq φpxq “ ErφpXptqqs it holds xµptq, f y “ xµp0q, f y ` ż t xµpsq, L f y ds

• r in some sense

B Bt µptq “ µptq L

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