Random Models of Dynamical Systems Introduction to SDEs (5/5) - PowerPoint PPT Presentation

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Random Models of Dynamical Systems Introduction to SDE’s (5/5) 4GM–AROM Fran¸ cois Le Gland INRIA Rennes + IRMAR http://www.irisa.fr/aspi/legland/insa-rennes/ December 10, 2018 1 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate 2 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate consider the simpler equation ż t ż t X p t q “ X p 0 q ` b p X p s qq ds ` σ p 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–independent coefficients: ‚ a d –dimensional drift vector b p x q defined on R d ‚ a d ˆ m diffusion matrix σ p x q defined on R d global Lipschitz condition: there exists a positive constant L ą 0 such that for any x , x 1 P R d | b p x q ´ b p x 1 q| ď L | x ´ x 1 | } σ p x q ´ σ p x 1 q} ď L | x ´ x 1 | and linear growth condition (simple consequence in this case): there exists a positive constant K ą 0 such that for any x P R d | b p x q| ď K p 1 ` | x |q and } σ p x q} ď K p 1 ` | x |q 3 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Strong vs. weak error objective: associated with a uniform subdivision 0 “ t 0 ă ¨ ¨ ¨ ă t k ă ¨ ¨ ¨ (with constant time–step h “ t k ´ t k ´ 1 ), design a numerical scheme ¯ X k that approximates the solution X p t k q , and provide an approximate continuous–time process ¯ X p t q (to be made precise later on) Definition the numerical scheme is strongly convergent of order α ą 0 if for any 0 ď t ď T X p t q| 2 u 1 { 2 ď C p T q h α t E | X p t q ´ ¯ Definition [approximation of moments] the numerical scheme is weakly convergent of order β ą 0 if for any regular enough real–valued function f and for any 0 ď t ď T | E r f p X p t qqs ´ E r f p ¯ X p t qqs | ď C p f , T q h β 4 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Remark if a numerical scheme is strongly convergent of order α ą 0, then it is also weakly convergent of the same order α ą 0 (for a Lipschitz continuous function f ) indeed: if | f p x q ´ f p x 1 q| ď L | x ´ x 1 | for any x , x 1 P R d , then | E r f p X p t qqs ´ E r f p ¯ X p t qqs | “ | E r f p X p t qq ´ f p ¯ X p t qqs | ď E | f p X p t qq ´ f p ¯ X p t qq| ď L E | X p t q ´ ¯ X p t q| ď L t E | X p t q ´ ¯ X p t q| 2 u 1 { 2 ď L C p T q h α 5 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Euler scheme special important case: Euler scheme same initial condition ¯ X 0 “ X p 0 q for k “ 0, and for any k ě 1 X k “ ¯ ¯ X k ´ 1 ` b p ¯ X k ´ 1 q p t k ´ t k ´ 1 q ` σ p ¯ X k ´ 1 q p B p t k q ´ B p t k ´ 1 qq and continuous–time approximation interpolating points ¯ X k at time instants t k X p t q “ ¯ ¯ X k ´ 1 ` b p ¯ X k ´ 1 q p t ´ t k ´ 1 q ` σ p ¯ X k ´ 1 q p B p t q ´ B p t k ´ 1 qq for any time t k ´ 1 ď t ď t k between two discretization times 6 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Euler approximation seen as an Itˆ o process, with frozen coefficients on each interval of the subdivision: indeed, for any t k ´ 1 ď t ď t k ż t ż t X p t q “ ¯ ¯ b p ¯ σ p ¯ X k ´ 1 ` X p π p s qqq ds ` X p π p s qqq dB p s q t k ´ 1 t k ´ 1 and more generally for any t ě 0 ż t ż t X p t q “ ¯ ¯ b p ¯ σ p ¯ X p 0 q ` X p π p s qqq ds ` X p π p s qqq dB p s q 0 0 where X p π p s qq “ ¯ ¯ π p s q “ t k ´ 1 and X k ´ 1 if t k ´ 1 ď s ă t k there exists a positive constant M p T q , independent of the time–step h , such that X p t q| 2 ď M p T q 0 ď t ď T E | ¯ max 7 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate 8 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Euler scheme: strong error estimate Theorem 1 the Euler scheme is strongly convergent of order 1 2 , i.e. X p t q| 2 “ O p h q 0 ď t ď T E | X p t q ´ ¯ max 9 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Proof for any time t k ´ 1 ď t ď t k between two discretization times, it holds ż t ż t X p t q “ X p t k ´ 1 q ` b p X p s qq ds ` σ p X p s qq dB p s q t k ´ 1 t k ´ 1 and (Euler approximation interpolating points ¯ X k at time instants t k ) X p t q “ ¯ ¯ X k ´ 1 ` b p ¯ X k ´ 1 q p t ´ t k ´ 1 q ` σ p ¯ X k ´ 1 q p B p t q ´ B p t k ´ 1 qq by difference, for any t k ´ 1 ď t ď t k ż t X p t q ´ ¯ X p t q “ X p t k ´ 1 q ´ ¯ r b p X p s qq ´ b p ¯ X k ´ 1 ` X k ´ 1 qs ds t k ´ 1 ż t r σ p X p s qq ´ σ p ¯ ` X k ´ 1 qs dB p s q t k ´ 1 10 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate using the Itˆ o formula yields X p t q| 2 “ | X p t k ´ 1 q ´ ¯ | X p t q ´ ¯ X k ´ 1 | 2 ż t X p s qq ˚ r b p X p s qq ´ b p ¯ p X p s q ´ ¯ ` 2 X k ´ 1 qs ds t k ´ 1 ż t X p s qq ˚ r σ p X p s qq ´ σ p ¯ p X p s q ´ ¯ ` 2 X k ´ 1 qs dB p s q t k ´ 1 ż t X k ´ 1 q} 2 ds } σ p X p s qq ´ σ p ¯ ` t k ´ 1 11 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate using the bound 2 u ˚ v ď | u | 2 ` | v | 2 , and taking expectation (assuming the stochastic integral is a (true, square–integrable) martingale), yields X p t q| 2 ď E | X p t k ´ 1 q ´ ¯ E | X p t q ´ ¯ X k ´ 1 | 2 ż t X p s q| 2 ds | X p s q ´ ¯ ` E t k ´ 1 ż t X k ´ 1 q| 2 ds | b p X p s qq ´ b p ¯ ` E t k ´ 1 ż t X k ´ 1 q} 2 ds } σ p X p s qq ´ σ p ¯ ` E t k ´ 1 12 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate note that | b p X p s qq ´ b p ¯ X k ´ 1 q| ď | b p X p s qq ´ b p X p t k ´ 1 qq| ` | b p X p t k ´ 1 qq ´ b p ¯ X k ´ 1 q| ď L r| X p s q ´ X p t k ´ 1 q| ` | X p t k ´ 1 q ´ ¯ X k ´ 1 |s and similarly } σ p X p s qq ´ σ p ¯ X k ´ 1 q} ď L r| X p s q ´ X p t k ´ 1 q| ` | X p t k ´ 1 q ´ ¯ X k ´ 1 |s with two different contributions to the error ‚ discretization error at previous iteration ‚ modulus of continuity of the solution 13 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate therefore X p t q| 2 ď p 1 ` 4 L 2 p t ´ t k ´ 1 qq E | X p t k ´ 1 q ´ ¯ E | X p t q ´ ¯ X k ´ 1 | 2 ż t ` 4 L 2 E | X p s q ´ X p t k ´ 1 q| 2 ds t k ´ 1 ż t X p s q| 2 ds | X p s q ´ ¯ ` E t k ´ 1 note that the modulus of continuity for the solution satisfies E | X p s q ´ X p t k ´ 1 q| 2 ď C p s ´ t k ´ 1 q hence X p t q| 2 ď p 1 ` 4 L 2 h q E | X p t k ´ 1 q ´ ¯ X k ´ 1 | 2 ` 4 L 2 C h 2 E | X p t q ´ ¯ ż t X p s q| 2 ds | X p s q ´ ¯ ` E t k ´ 1 14 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate the Gronwall lemma yields X p t q| 2 ď E | X p t q ´ ¯ X k ´ 1 | 2 ` 4 L 2 C h 2 s exp t t ´ t k ´ 1 u ď rp 1 ` 4 L 2 h q E | X p t k ´ 1 q ´ ¯ introducing t k ´ 1 ď t ď t k E | X p t q ´ ¯ X p t q| 2 ε k “ max it holds ε k ď p 1 ` 4 L 2 h q exp t h u ε k ´ 1 ` 4 L 2 C h 2 exp t h u and by induction 4 L 2 C h 2 exp t h u p 1 ` 4 L 2 h q exp t h u ´ 1 rp 1 ` 4 L 2 h q exp t h us k ε k ď note that 4 L 2 C h 2 exp t h u 4 L 2 C h 2 p 1 ` 4 L 2 h q exp t h u ´ 1 “ 4 L 2 h ` p 1 ´ exp t´ h uq “ O p h q 15 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate for any k “ 1 ¨ ¨ ¨ t T { h u , the following bound holds rp 1 ` 4 L 2 h q exp t h us k ď rp 1 ` 4 L 2 h q exp t h us t T { h u ď exp tp 1 ` 4 L 2 q T u therefore X p t q| 2 “ 0 ď t ď T E | X p t q ´ ¯ max k “ 1 ¨¨¨ t T { h u ε k max 4 L 2 C h 2 4 L 2 h ` p 1 ´ exp t´ h uq exp tp 1 ` 4 L 2 q T u “ O p h q ď l 16 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate 17 / 37

Stochastic differential equations Euler scheme: strong error estimate Euler scheme: weak error estimate Euler scheme: weak error estimate let T ě 0 be fixed (as in the PDE) and consider specifically a uniform subdivision of the form 0 “ t 0 ă ¨ ¨ ¨ ă t k ă ¨ ¨ ¨ ă t n “ T of the interval r 0 , T s (with constant time–step h “ T { n ) Theorem 2 under some additional technical assumptions (on the coefficients of the SDE and on the test function) the Euler scheme is weakly convergent of order 1, i.e. | E r f p X p T qqs ´ E r f p ¯ X p T qqs | “ O p h q even more E r f p X p T qqs ´ E r f p ¯ X p T qqs “ C p f , T q h ` O p h 2 q 18 / 37