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 1 / 50
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 2 / 50
Stochastic differential equations Solution as a Markov process Definition, assumptions on the coefficients consider the equation ż t ż t X p t q “ X p 0 q ` b p s , X p s qq ds ` σ p s , X p s qq dB p s q 0 0 with a m –dimensional Brownian motion B “ p B p t q , t ě 0 q , and time–dependent coefficients: ‚ a d –dimensional drift vector b p t , x q defined on r 0 , 8q ˆ R d ‚ a d ˆ m diffusion matrix σ p t , x q defined on r 0 , 8q ˆ R d global Lipschitz condition: there exists a positive constant L ą 0 such that for any t ě 0 and any x , x 1 P R d | b p t , x q´ b p t , x 1 q| ď L | x ´ x 1 | } σ p t , x q´ σ p t , x 1 q} ď L | x ´ x 1 | and linear growth condition: there exists a positive constant K ą 0 such that for any t ě 0 and any x P R d | b p t , x q| ď K p 1 ` | x |q and } σ p t , x q} ď K p 1 ` | x |q 2 / 50
Stochastic differential equations Solution as a Markov process a solution to the SDE is any process X in M 2 pr 0 , T sq such that the identity holds almost surely the condition that X is in M 2 pr 0 , T sq makes sure that the stochastic integral ż t σ p s , X p s qq dB p s q 0 defines a (true, square–integrable) martingale: indeed, the vector–valued stochastic integral makes sense iff for any v P R d , the one–dimensional stochastic integral ż t ż t v ˚ σ p s , X p s qq dB p s q v ˚ σ p s , X p s qq dB p s q “ 0 0 makes sense, i.e. iff ż T } σ p s , X p s qq σ ˚ p s , X p s qq} ds ă 8 E 0 and note that ż T ż T } σ p s , X p s qq σ ˚ p s , X p s qq} ds ď 2 K 2 E p 1 ` | X p s q| 2 q ds E 0 0 3 / 50
Stochastic differential equations Solution as a Markov process Lemma [Gronwall lemma] if the nonnegative function u p t q satisfies the functional relation: for any t ě 0 and for some nonnegative constants a , c ě 0 ż t u p t q ď a ` c u p s q ds 0 then for any t ě 0 u p t q ď a exp t c t u Proof assume c ą 0 without loss of generality, and note that ż t ż t d dt r exp t´ c t u u p s q ds s “ exp t´ c t u r u p t q´ c u p s q ds s ď a exp t´ c t u 0 0 integration yields ż t ż t exp t´ c s u ds “ a exp t´ c t u u p s q ds ď a c p 1 ´ exp t´ c t uq 0 0 hence ż t u p s q ds ď a c p exp t c t u ´ 1 q l 0 4 / 50
Stochastic differential equations Solution as a Markov process Lemma [sequential Gronwall lemma] if the nonnegative functions u n p t q satisfies the functional relation: for any t ě 0 and any n ě 1 and for some nonnegative constants a , c ě 0 ż t u n p t q ď a ` c u n ´ 1 p s q ds with u 0 p t q ” ¯ u 0 then for any t ě 0 and any n ě 1 u n p t q ď a exp t c t u ` p c t q n u ¯ n ! Remark the following uniform estimate holds p c t q n max n ě 1 u n p t q ď a exp t c t u ` r max s ¯ u n ! n ě 1 and asymptotically lim sup n Ñ8 u n p t q ď a exp t c t u 5 / 50
Stochastic differential equations Solution as a Markov process Proof actually, the following stronger estimate is proved by induction u n p t q ď a r 1 ` c t ` ¨ ¨ ¨ ` p c t q n ´ 1 p n ´ 1 q ! s ` p c t q n u ď a exp t c t u ` p c t q n ¯ ¯ u n ! n ! clearly, the estimate holds for n “ 1 assuming that the estimate holds at stage p n ´ 1 q , then ż t u n p t q “ a ` c u n ´ 1 p s q ds 0 ż t ż t r 1 ` c s ` ¨ ¨ ¨ ` p c s q n ´ 2 p c s q n ´ 1 ď a r 1 ` c p n ´ 2 q ! s ds s ` r c p n ´ 1 q ! ds s ¯ u 0 0 “ a r 1 ` c t ` ¨ ¨ ¨ ` p c t q n ´ 1 p n ´ 1 q ! s ` p c t q n u ¯ n ! i.e. the estimate holds at stage n l 6 / 50
Stochastic differential equations Solution as a Markov process simple (yet useful) formula ż t ż t ψ p s q ds | p ď t p ´ 1 | ψ p s q| p ds | 0 0 hence (taking ψ p s q “ φ 2 p s q and using p { 2 in place of p ) ż t ż t | φ p s q| 2 ds q p { 2 ď t p { 2 ´ 1 | φ p s q| p ds p 0 0 older inequality for conjugate exponents p , p 1 yields Proof using the H¨ ż t ż t ż t 1 p 1 ds q 1 { p 1 p | ψ p s q| p ds q 1 { p | ψ p s q ds | ď p 0 0 0 and note that p { p 1 “ p ´ 1 l 7 / 50
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 X p 0 q , there exists a unique solution to the SDE ż t ż t X p t q “ X p 0 q ` b p s , X p s qq ds ` σ p s , X p s qq dB p s q 0 0 Proof uniqueness: let X “ p X p t q , t ě 0 q and X 1 “ p X 1 p t q , t ě 0 q be two solutions, with the same initial condition X p 0 q “ X 1 p 0 q by difference, for any 0 ď t ď T ż t | X p t q ´ X 1 p t q| ď | b p s , X p s qq ´ b p s , X 1 p s qq ds | 0 ż t p σ p s , X p s qq ´ σ p s , X 1 p s qqq dB p s q | ` | 0 8 / 50
Stochastic differential equations Solution as a Markov process hence ż t E | X p t q ´ X 1 p t q| 2 ď 2 E | p b p s , X p s qq ´ b p s , X 1 p s qqq ds | 2 0 ż t p σ p s , X p s qq ´ σ p s , X 1 p s qqq dB p s q | 2 ` 2 E | 0 ż t | b p s , X p s qq ´ b p s , X 1 p s qq| 2 ds ď 2 t E 0 ż t } σ p s , X p s qq ´ σ p s , X 1 p s qq} 2 ds ` 2 E 0 ż t ď 2 L 2 p T ` 1 q E | X p s q ´ X 1 p s q| 2 ds 0 it follows from the Gronwall lemma that for any 0 ď t ď T E | X p t q ´ X 1 p t q| 2 “ 0 9 / 50
Stochastic differential equations Solution as a Markov process Picard iteration: for n “ 0, let X 0 p t q ” X p 0 q for any 0 ď t ď T , and for any n ě 1 consider the Itˆ o process ż t ż t X n p t q “ X p 0 q ` b p s , X n ´ 1 p s qq ds ` σ p s , X n ´ 1 p s qq dB p s q 0 0 no localization is needed here, thanks to the following a priori estimate: there exists a positive constant M p T q such that for any n ě 1 E | X n p t q| 2 ď M p T q sup ( ‹ ) 0 ď t ď T clearly, the estimate holds for n “ 0, and by induction if the estimate holds at stage p n ´ 1 q , then ż t ż t p 1 ` | X n ´ 1 p s q|q 2 ds } σ p s , X n ´ 1 p s qq σ ˚ p s , X n ´ 1 p s qq} ds ď K 2 E E 0 0 ż t ď 2 K 2 p t ` E | X n ´ 1 p s q| 2 ds q 0 in other words: the integrand s ÞÑ σ p s , X n ´ 1 p s qq belongs to M 2 pr 0 , T sq 10 / 50
Stochastic differential equations Solution as a Markov process a priori estimate: ż t ż t | X n p t q| ď | X p 0 q| ` | b p s , X n ´ 1 p s qq ds | ` | σ p s , X n ´ 1 p s qq dB p s q| 0 0 and E | X n p t q| 2 ´ 3 E | X p 0 q| 2 ż t ż t b p s , X n ´ 1 p s qq ds | 2 ` 3 E | σ p s , X n ´ 1 p s qq dB p s q| 2 ď 3 E | 0 0 ż t ż t | b p s , X n ´ 1 p s qq| 2 ds ` 3 E } σ p s , X n ´ 1 p s qq} 2 ds ď 3 t E 0 0 ż t ż t ď 6 K 2 t E p 1 ` | X n ´ 1 p s q| 2 q ds ` 6 K 2 E p 1 ` | X n ´ 1 p s q| 2 q ds 0 0 ż t ď 6 K 2 T p T ` 1 q ` 6 K 2 p T ` 1 q E | X n ´ 1 p s q| 2 ds 0 11 / 50
Stochastic differential equations Solution as a Markov process in other words, the sequence u n p t q “ E | X n p t q| 2 satisfies the functional relation ż t u 0 p t q ” E | X p 0 q| 2 u n p t q ď a p T q ` c p T q u n ´ 1 p s q ds with 0 it follows from the sequential Gronwall lemma that E | X n p t q| 2 ď a p T q exp t c p T q T u ` p c p T q T q n E | X p 0 q| 2 sup n ! 0 ď t ď T which proves the a priori estimate ( ‹ ) where the bound n ě 1 rp c p T q T q n s E | X p 0 q| 2 M p T q “ a p T q exp t c p T q T u ` max n ! depends on T , K and E | X p 0 q| 2 , and does not depend on L 12 / 50
Stochastic differential equations Solution as a Markov process uniform a priori estimate: ż s ż s | X n p s q| ď | X p 0 q| ` | b p u , X n ´ 1 p u qq du | ` | σ p u , X n ´ 1 p u qq dB p u q| 0 0 uniform upper bound ż s sup | X n p s q| ď | X p 0 q| ` sup | b p u , X n ´ 1 p u qq du | 0 ď s ď t 0 ď s ď t 0 ż s ` sup | σ p u , X n ´ 1 p u qq dB p u q| 0 ď s ď t 0 13 / 50
Stochastic differential equations Solution as a Markov process using the Doob inequality yields | X n p s q| 2 s ´ 3 E | X p 0 q| 2 E r sup 0 ď s ď t ż s ż s b p u , X n ´ 1 p u qq du | 2 s ` 3 E r sup σ p u , X n ´ 1 p u qq dB p u q| 2 s ď 3 E r sup | | 0 ď s ď t 0 ď s ď t 0 0 ż s ż t | b p u , X n ´ 1 p u qq| 2 du s ` 12 E } σ p s , X n ´ 1 p s qq} 2 ds ď 3 t E r sup 0 ď s ď t 0 0 ż t ż t ď 6 K 2 t E p 1 ` | X n ´ 1 p s q| 2 q ds ` 24 K 2 E p 1 ` | X n ´ 1 p s q| 2 q ds 0 0 ż t ď 6 K 2 T p T ` 4 q ` 6 K 2 p T ` 4 q | X n ´ 1 p u q| 2 s ds E r sup 0 ď u ď s 0 14 / 50
Stochastic differential equations Solution as a Markov process in other words, the sequence | X n p s q| 2 s u n p t q “ E r sup ¯ 0 ď s ď t satisfies the functional relation ż t u 0 p t q ” E | X p 0 q| 2 u n p t q ď ¯ ¯ a p T q ` ¯ c p T q u n ´ 1 p s q ds ¯ with ¯ 0 it follows from the sequential Gronwall lemma that the stronger uniform a priori estimate holds c p T q T q n n ě 1 rp ¯ | X n p t q| 2 s ď ¯ s E | X p 0 q| 2 E r sup a p T q exp t ¯ c p T q T u ` max n ! 0 ď t ď T where the bound depends on T , K and E | X p 0 q| 2 , and does not depend on L 15 / 50
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