NUMERICS OF STOCHASTIC SYSTEMS WITH MEMORY

Transcription

1 NUMERICS OF STOCHASTIC SYSTEMS WITH MEMORY Salah Mohammed a Institut Mittag-Leffler Royal Swedish Academy of Sciences Sweden: December 13, 2007 NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.1/95

2 Acknowledgment Joint work with E. Buckwar, R. Kuske and T. Shardlow. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.2/95

3 Acknowledgment Joint work with E. Buckwar, R. Kuske and T. Shardlow. Research supported by NSF Grants DMS , DMS , DMS , DMS and Canadian PIMS. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.2/95

4 Outline Objective: To approximate functions of solutions of stochastic delay differential equations (sdde s) arising in applications. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.3/95

5 Outline Objective: To approximate functions of solutions of stochastic delay differential equations (sdde s) arising in applications. Formulation of the Euler scheme for sdde s. Convergence rate of order 1. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.3/95

6 Outline Objective: To approximate functions of solutions of stochastic delay differential equations (sdde s) arising in applications. Formulation of the Euler scheme for sdde s. Convergence rate of order 1. For a semimartingale x, must study increments of ψ(x(t τ), x(t)) across partition points. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.3/95

7 Outline Objective: To approximate functions of solutions of stochastic delay differential equations (sdde s) arising in applications. Formulation of the Euler scheme for sdde s. Convergence rate of order 1. For a semimartingale x, must study increments of ψ(x(t τ), x(t)) across partition points. No semimartingale properties for processes of the form (x(t τ), x(t)). But need an Itô formula! NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.3/95

8 Outline Objective: To approximate functions of solutions of stochastic delay differential equations (sdde s) arising in applications. Formulation of the Euler scheme for sdde s. Convergence rate of order 1. For a semimartingale x, must study increments of ψ(x(t τ), x(t)) across partition points. No semimartingale properties for processes of the form (x(t τ), x(t)). But need an Itô formula! Get a tame Itô formula for ψ(x(t τ), x(t)). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.3/95

9 Outline-contd Proof of the Euler scheme: Uses tame Itô formula, tame character and Fréchet differentiability of the Euler approximation in the initial path, estimates on the Malliavin derivatives of the solution, Malliavin and Fréchet derivatives of the Euler approximation. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.4/95

10 Outline-contd Proof of the Euler scheme: Uses tame Itô formula, tame character and Fréchet differentiability of the Euler approximation in the initial path, estimates on the Malliavin derivatives of the solution, Malliavin and Fréchet derivatives of the Euler approximation. Set-up is non-anticipating, but proof of convergence requires anticipating stochastic calculus techniques. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.4/95

11 Outline-contd Proof of the Euler scheme: Uses tame Itô formula, tame character and Fréchet differentiability of the Euler approximation in the initial path, estimates on the Malliavin derivatives of the solution, Malliavin and Fréchet derivatives of the Euler approximation. Set-up is non-anticipating, but proof of convergence requires anticipating stochastic calculus techniques. Implementation of the scheme does not require knowledge of the Malliavin calculus. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.4/95

12 Outline-contd Proof of the Euler scheme: Uses tame Itô formula, tame character and Fréchet differentiability of the Euler approximation in the initial path, estimates on the Malliavin derivatives of the solution, Malliavin and Fréchet derivatives of the Euler approximation. Set-up is non-anticipating, but proof of convergence requires anticipating stochastic calculus techniques. Implementation of the scheme does not require knowledge of the Malliavin calculus. Unlike (sode s), sdde s do not correspond to diffusions on Euclidean space. Thus techniques from deterministic pde s do not apply. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.4/95

13 Motivation Sdde s model noisy physical processes with memory: Laser dynamics with delayed feedback (Buldú, etal (2001), and Masoller (2002, 2003)). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.5/95

14 Motivation Sdde s model noisy physical processes with memory: Laser dynamics with delayed feedback (Buldú, etal (2001), and Masoller (2002, 2003)). Stochastic oscillator ensembles with delayed coupling (Goldobin, Rosenblum, Pikovsky (2003)). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.5/95

15 Motivation Sdde s model noisy physical processes with memory: Laser dynamics with delayed feedback (Buldú, etal (2001), and Masoller (2002, 2003)). Stochastic oscillator ensembles with delayed coupling (Goldobin, Rosenblum, Pikovsky (2003)). Models of delayed visual feedback systems (Beuter and Vasilakos (1993)), and (Longtin, Milton, Bos and Mackey (1990)). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.5/95

16 Motivation Sdde s model noisy physical processes with memory: Laser dynamics with delayed feedback (Buldú, etal (2001), and Masoller (2002, 2003)). Stochastic oscillator ensembles with delayed coupling (Goldobin, Rosenblum, Pikovsky (2003)). Models of delayed visual feedback systems (Beuter and Vasilakos (1993)), and (Longtin, Milton, Bos and Mackey (1990)). Option-pricing models with memory (Arriojas, Hu, Mohammed and Pap (2007)). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.5/95

17 Motivation-Contd Model equations are non-linear and do not allow for explicit solutions. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.6/95

18 Motivation-Contd Model equations are non-linear and do not allow for explicit solutions. Hence need numerical approximation methods of solution: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.6/95

19 Sfde approximations NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.7/95

20 Sfde approximations Strong (or almost sure) Euler scheme (order 1/2) and strong Milstein scheme (order 1) for sdde s were developed by Ahmed, Elsanousi and Mohammed [A], Mohammed [Mo.1], Hu, Mohammed and Yan [H.M.Y] and Baker and Buckwar [B.B], Küchler and Platen [Kü.P]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.7/95

21 Sfde approximations Strong (or almost sure) Euler scheme (order 1/2) and strong Milstein scheme (order 1) for sdde s were developed by Ahmed, Elsanousi and Mohammed [A], Mohammed [Mo.1], Hu, Mohammed and Yan [H.M.Y] and Baker and Buckwar [B.B], Küchler and Platen [Kü.P]. Weak approximations for sode s (without memory) are well-developed (Bally and Talay [B.T], Kloeden and Platen [K.P], Kohatsu-Higa [K]). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.7/95

22 Sfde approximations Strong (or almost sure) Euler scheme (order 1/2) and strong Milstein scheme (order 1) for sdde s were developed by Ahmed, Elsanousi and Mohammed [A], Mohammed [Mo.1], Hu, Mohammed and Yan [H.M.Y] and Baker and Buckwar [B.B], Küchler and Platen [Kü.P]. Weak approximations for sode s (without memory) are well-developed (Bally and Talay [B.T], Kloeden and Platen [K.P], Kohatsu-Higa [K]). Weak Euler scheme of order 1 for semilinear sfde s: Buckwar and Shardlow (2005); linear smooth memory drift term; memoryless diffusion term; 1-dim l noise: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.7/95

23 Approximations cont d x(t) = v + t 0 0 τ t + f(x(u)) du + 0 η(t), τ < t < 0. x(u + s) µ(s) ds du t 0 g(x(u)) dw (u), t 0 Initial condition (v, η) M 2 := R d L 2 ([ τ, 0], R d ). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.8/95

24 Approximations cont d x(t) = v + t 0 0 τ t + f(x(u)) du + 0 η(t), τ < t < 0. x(u + s) µ(s) ds du t 0 g(x(u)) dw (u), t 0 Initial condition (v, η) M 2 := R d L 2 ([ τ, 0], R d ). Embed sfde as a semilinear see (without memory) in Hilbert space M 2. Weak approximation in M 2. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.8/95

25 Approximations cont d x(t) = v + t 0 0 τ t + f(x(u)) du + 0 η(t), τ < t < 0. x(u + s) µ(s) ds du t 0 g(x(u)) dw (u), t 0 Initial condition (v, η) M 2 := R d L 2 ([ τ, 0], R d ). Embed sfde as a semilinear see (without memory) in Hilbert space M 2. Weak approximation in M 2. Duality methods for weak Euler scheme-independently by Clément, Kohatsu-Higa and Lamberton [CK-HL]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.8/95

26 Approximations cont d In this talk, we prove weak convergence of order 1 of the Euler scheme for fully non-linear sdde s. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.9/95

27 Approximations cont d In this talk, we prove weak convergence of order 1 of the Euler scheme for fully non-linear sdde s. Allow for: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.9/95

28 Approximations cont d In this talk, we prove weak convergence of order 1 of the Euler scheme for fully non-linear sdde s. Allow for: multiple discrete delays NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.9/95

29 Approximations cont d In this talk, we prove weak convergence of order 1 of the Euler scheme for fully non-linear sdde s. Allow for: multiple discrete delays smooth memory NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.9/95

30 Approximations cont d In this talk, we prove weak convergence of order 1 of the Euler scheme for fully non-linear sdde s. Allow for: multiple discrete delays smooth memory multidimensional Brownian noise NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.9/95

31 The weak Euler scheme NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.10/95

32 The weak Euler scheme The SDDE: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.10/95

33 The weak Euler scheme The SDDE: x(t) = η(0) + + t σ t σ f ( x(u τ 1 ), x(u) ) du g ( x(u τ 2 ), x(u) ) dw (u), t σ, η(t σ), σ τ t < σ, τ := τ 1 τ 2. (I) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.10/95

34 The weak Euler scheme The SDDE: t η(0) + f ( x(u τ 1 ), x(u) ) du σ x(t) = t + g ( x(u τ 2 ), x(u) ) dw (u), t σ, σ η(t σ), σ τ t < σ, τ := τ 1 τ 2. (I) Coefficients f, g : R 2 R are of class Cb 3. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.10/95

35 The weak Euler scheme The SDDE: x(t) = η(0) + + t σ t f ( x(u τ 1 ), x(u) ) du g ( x(u τ 2 ), x(u) ) dw (u), t σ, σ η(t σ), σ τ t < σ, τ := τ 1 τ 2. (I) Coefficients f, g : R 2 R are of class Cb 3. Initial (random) path η C 1 ([ τ, 0], R). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.10/95

36 The weak Euler scheme The SDDE: x(t) = η(0) + + t σ t f ( x(u τ 1 ), x(u) ) du g ( x(u τ 2 ), x(u) ) dw (u), t σ, σ η(t σ), σ τ t < σ, τ := τ 1 τ 2. (I) Coefficients f, g : R 2 R are of class Cb 3. Initial (random) path η C 1 ([ τ, 0], R). Initial instant σ 0. (< ) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.10/95

37 The weak Euler scheme cont d Noise: One-dimensional Brownian motion W on a filtered probability space (Ω, F, (F t ) t 0, P ). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.11/95

38 The weak Euler scheme cont d Noise: One-dimensional Brownian motion W on a filtered probability space (Ω, F, (F t ) t 0, P ). Unique solution x := x( ; σ, η) : [σ τ, a] Ω R of (I), fixed a > σ. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.11/95

39 The weak Euler scheme cont d Noise: One-dimensional Brownian motion W on a filtered probability space (Ω, F, (F t ) t 0, P ). Unique solution x := x( ; σ, η) : [σ τ, a] Ω R of (I), fixed a > σ. Partition π := { τ = t m < t m+1 < < t 1 < 0 = t 0 < t 1 < t 2 < t n 1 < t n = a} of [ τ, a], with mesh: π := max{(t i t i 1 ) : m + 1 i n}. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.11/95

40 The weak Euler scheme cont d Noise: One-dimensional Brownian motion W on a filtered probability space (Ω, F, (F t ) t 0, P ). Unique solution x := x( ; σ, η) : [σ τ, a] Ω R of (I), fixed a > σ. Partition π := { τ = t m < t m+1 < < t 1 < 0 = t 0 < t 1 < t 2 < t n 1 < t n = a} of [ τ, a], with mesh: π := max{(t i t i 1 ) : m + 1 i n}. For any u [σ, a], define u := t i 1 σ whenever u [t i 1, t i ] [σ, a]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.11/95

41 The weak Euler scheme cont d NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.12/95

42 The weak Euler scheme cont d Euler approximation: y := y( ; σ, η) : [σ τ, a] Ω R of x( ; σ, η) is the solution of the sdde: y(t) = η(0) + t t σ f ( y ( u τ 1 ), y ( u )) du + g ( y ( ) ( )) u τ 2, y u dw (u), t σ, σ η(t σ), σ τ t < σ, τ := τ 1 τ 2. (II) ( η π ti s ) ( s ti 1 ) (s) := η(t i 1 ) + η(t i ), t i t i 1 t i t i 1 s [t i 1, t i ), m + 1 i 0. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.12/95

43 The weak Euler scheme cont d Main result: Weak convergence of order 1 for the Euler scheme of the sdde (I). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.13/95

44 Theorem 1-Weak Convergence Let π be a partition of [ τ, a] with mesh π, and φ : R R be of class C 3 b. that the coefficients f, g are C 3 In the sdde (I), assume b. Let x( ; σ, η) be the unique solution of (I) starting at σ (0, a] with initial path η C 1( [ τ, 0], R ). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.14/95

45 Theorem 1-Weak Convergence Let π be a partition of [ τ, a] with mesh π, and φ : R R be of class C 3 b. that the coefficients f, g are C 3 In the sdde (I), assume b. Let x( ; σ, η) be the unique solution of (I) starting at σ (0, a] with initial path η C 1( [ τ, 0], R ). Denote by y( ; σ, η) the Euler approximation to x( ; σ, η) defined by (II). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.14/95

46 Theorem 1-Weak Convergence Let π be a partition of [ τ, a] with mesh π, and φ : R R be of class C 3 b. that the coefficients f, g are C 3 In the sdde (I), assume b. Let x( ; σ, η) be the unique solution of (I) starting at σ (0, a] with initial path η C 1( [ τ, 0], R ). Denote by y( ; σ, η) the Euler approximation to x( ; σ, η) defined by (II). Then there is a positive constant C and a positive integer q such that NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.14/95

47 Theorem 1 cont d Eφ(x(t; σ, η)) Eφ(y(t; σ, η π )) C(1 + η q C 1 ) π for all η C 1( [ τ, 0], R ) and all t [σ τ, a]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.15/95

48 Theorem 1 cont d Eφ(x(t; σ, η)) Eφ(y(t; σ, η π )) C(1 + η q C ) π 1 for all η C 1( [ τ, 0], R ) and all t [σ τ, a]. The constant C may depend on a, q and the test function φ, but is independent of π, η, σ [0, a] and t [σ τ, a]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.15/95

49 Markov Property For the solution x : [ τ, a] Ω R of sdde (I), denote its segment x t C([ τ, 0], R), t [0, a], by x t (s) := x(t + s), s [ τ, 0], t [0, a]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.16/95

50 Markov Property For the solution x : [ τ, a] Ω R of sdde (I), denote its segment x t C([ τ, 0], R), t [0, a], by x t (s) := x(t + s), s [ τ, 0], t [0, a]. x t C([ τ, 0], R), t 0, is Markov. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.16/95

51 The segment process η x t x(t) τ t τ 0 t a NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.17/95

52 Outline of Proof Step 1: Let t [σ, a] and π := {t 0, t 1, t 2,, t n } be a partition of [0, a]. W.l.o.g, assume that σ = t 0 = 0, t = t n. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.18/95

53 Outline of Proof Step 1: Let t [σ, a] and π := {t 0, t 1, t 2,, t n } be a partition of [0, a]. W.l.o.g, assume that σ = t 0 = 0, t = t n. Using telescoping, the Markov property for x t and y t, and Fréchet differentiability of y(t n ; t i, η) in η: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.18/95

54 Outline of Proof Step 1: Let t [σ, a] and π := {t 0, t 1, t 2,, t n } be a partition of [0, a]. W.l.o.g, assume that σ = t 0 = 0, t = t n. Using telescoping, the Markov property for x t and y t, and Fréchet differentiability of y(t n ; t i, η) in η: Eφ ( x(t n ; 0, η) ) Eφ ( y(t n ; 0, η) ) = Eφ ( y(t n ; t n, x tn ( ; 0, η)) ) Eφ ( y(t n ; 0, η) ) n = {Eφ ( y(t n ; t i, x ti ( ; 0, η)) ) i=1 Eφ ( y(t n ; t i 1, x ti 1 ( ; 0, η)) ) } NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.18/95

55 Outline of Proof cont d = = n i=1 n i=1 { ( Eφ y(tn ; t i, x ti ( ; t i 1, x ti 1 ( ; 0, η)))) E Eφ ( y(t n ; t i, y ti ( ; t i 1, x ti 1 ( ; 0, η))) )} 1 0 D(φ y) ( t n ; t i, λx ti ( ; t i 1, x ti 1 ( ; 0, η)) + (1 λ)y ti ( ; ti 1, x ti 1 ( ; 0, η)) ) dλ [x ti ( ; ti 1, x ti 1 ( ; 0, η) ) y ti ( ; ti 1, x ti 1 ( ; 0, η) )]. (M.V. Theorem) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.19/95

56 Outline of Proof cont d Step 2: Main task is to show that each of the terms in the above sum is O((t i t i 1 ) 2 ): Use the tame Itô formula. Get multiple Skorohod integrals of the form NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.20/95

57 Outline of Proof cont d Step 2: Main task is to show that each of the terms in the above sum is O((t i t i 1 ) 2 ): Use the tame Itô formula. Get multiple Skorohod integrals of the form J i 1 := 0 J i 2 := 0 J i 3 := t i 1 t i Y (ds) t i 1 t i Y (ds) 0 t i 1 t i Y (ds) ti +s t i 1 ti +s t i 1 ti +s t i 1 u t i 1 Σ 1 (v) dv dw (u), u t i 1 Σ 2 (v) dw (v τ 2 ) dw (u), u t i 1 Σ 3 (v) dw (v τ 1 ) du. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.20/95

58 Outline of Proof cont d The discrete random measure Y and the processes Σ j, j = 1, 2, 3, are Malliavin smooth and possibly anticipate the lagged Brownian motions W ( τ i ), i = 1, 2. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.21/95

59 Outline of Proof cont d The discrete random measure Y and the processes Σ j, j = 1, 2, 3, are Malliavin smooth and possibly anticipate the lagged Brownian motions W ( τ i ), i = 1, 2. Step 3: To estimate the expectations EJj i in Step 2, use the definition of the Skorohod integral as adjoint of the weak differentiation operator, coupled with estimates on higher-order moments of Malliavin derivatives of Σ j s, j = 1, 2, 3. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.21/95

60 Outline of Proof cont d The discrete random measure Y and the processes Σ j, j = 1, 2, 3, are Malliavin smooth and possibly anticipate the lagged Brownian motions W ( τ i ), i = 1, 2. Step 3: To estimate the expectations EJj i in Step 2, use the definition of the Skorohod integral as adjoint of the weak differentiation operator, coupled with estimates on higher-order moments of Malliavin derivatives of Σ j s, j = 1, 2, 3. These estimates follow from higher moments of the solution x, its Euler approximations y and Malliavin derivatives of linearizations of y. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.21/95

61 Outline of Proof cont d This gives EJ i j = O((t i t i 1 ) 2 ), j = 1, 2, 3. Summing over i = 1,, n, we get the required order of convergence 1 for the weak Euler scheme. Step 4: Replace η in y(t; σ, η) by its P-L approx η π via the estimates Eφ(x(t; σ, η)) Eφ(x(t; σ, η π )) C η η π C C η π. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.22/95

62 THE PROOF NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.23/95

63 Example NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.24/95

64 Example One-dimensional sdde: dx(t) = g(x(t 1), X(t)) dw (t), t > 0, X(t) = W (t), 1 t 0. g : R 2 R smooth function. For Euler scheme of order 1, seek a stochastic differential of g(x(t 1), X(t)) on RHS of sdde. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.24/95

65 Example Cont d For t (0, 1], formally expect something like: dg(x(t 1), X(t)) = g x 2 (W (t 1), X(t)) g(w (t 1), X(t)) dw (t) + g x 1 (W (t 1), X(t)) dw (t 1) (anticipating!) + second-order terms NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.25/95

66 Example Cont d LHS is adapted but anticipating integral(s) on RHS. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.26/95

67 Example Cont d LHS is adapted but anticipating integral(s) on RHS. (F t ) 0 t 1 -adapted process (X(t 1), X(t)) R 2 is not a semimartingale with respect to any natural filtration. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.26/95

68 Example Cont d LHS is adapted but anticipating integral(s) on RHS. (F t ) 0 t 1 -adapted process (X(t 1), X(t)) R 2 is not a semimartingale with respect to any natural filtration. Still need an Itô formula for tame functions: g(x(t 1), X(t)) = g(x t ( 1), X t (0)). where X t (s) := X(t + s), s [ 1, 0], t 0. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.26/95

69 The tame Itô formula NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.27/95

70 The tame Itô formula Objective is to obtain an Itô formula for tame functionals on the Banach C([ τ, 0], R d ), acting on segments of sample-continuous random processes [ τ, ] Ω R d : tame Itô formula ([H.M.Y]) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.27/95

71 The tame Itô formula Objective is to obtain an Itô formula for tame functionals on the Banach C([ τ, 0], R d ), acting on segments of sample-continuous random processes [ τ, ] Ω R d : tame Itô formula ([H.M.Y]) First, some notation: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.27/95

72 The tame Itô formula Objective is to obtain an Itô formula for tame functionals on the Banach C([ τ, 0], R d ), acting on segments of sample-continuous random processes [ τ, ] Ω R d : tame Itô formula ([H.M.Y]) First, some notation: W (t), t 0, one-dimensional standard Brownian motion on a filtered probability space (Ω, F, (F t ) t 0, P ). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.27/95

73 The tame Itô formula Objective is to obtain an Itô formula for tame functionals on the Banach C([ τ, 0], R d ), acting on segments of sample-continuous random processes [ τ, ] Ω R d : tame Itô formula ([H.M.Y]) First, some notation: W (t), t 0, one-dimensional standard Brownian motion on a filtered probability space (Ω, F, (F t ) t 0, P ). For simplicity, set W (t) := 0 if t 0. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.27/95

74 The tame Itô formula Objective is to obtain an Itô formula for tame functionals on the Banach C([ τ, 0], R d ), acting on segments of sample-continuous random processes [ τ, ] Ω R d : tame Itô formula ([H.M.Y]) First, some notation: W (t), t 0, one-dimensional standard Brownian motion on a filtered probability space (Ω, F, (F t ) t 0, P ). For simplicity, set W (t) := 0 if t 0. D := the weak (Malliavin) differentiation operator associated with {W (t) : t 0}. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.27/95

75 The tame Itô formula Cont d Let p > 1, k a positive integer; L k,p := L p ([0, a], D k,p ), where D k,p is the closure of all random variables Y with k-th weak derivatives in L p (Ω, H k ) under the norm Y k,p := (E Y p ) 1/p + ( k j=1 E D j Y p H j ) 1/p. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.28/95

76 The tame Itô formula Cont d Let p > 1, k a positive integer; L k,p := L p ([0, a], D k,p ), where D k,p is the closure of all random variables Y with k-th weak derivatives in L p (Ω, H k ) under the norm Y k,p := (E Y p ) 1/p + ( k j=1 E D j Y p H j ) 1/p. In above formula, H := L 2 ([0, a], R). The spaces L k,p loc, p > 4, are defined to be the set of all processes X such that there is an increasing sequence of F-measurable sets A n, n 1, and processes X n L k,p, NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.28/95

77 The tame Itô formula Cont d n 1, such that X = X n a.s. on A n for each n 1, and A n = Ω. Weak differentiation operator D is local and n=1 hence extends unambiguously to the spaces L k,p loc, p > 4. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.29/95

78 The tame Itô formula Cont d n 1, such that X = X n a.s. on A n for each n 1, and A n = Ω. Weak differentiation operator D is local and n=1 hence extends unambiguously to the spaces L k,p loc, p > 4. See ([Nu.1], pp. 61, 151, 161) for further properties of weak derivatives and the spaces L k,p. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.29/95

79 The tame Itô formula Cont d n 1, such that X = X n a.s. on A n for each n 1, and A n = Ω. Weak differentiation operator D is local and n=1 hence extends unambiguously to the spaces L k,p loc, p > 4. See ([Nu.1], pp. 61, 151, 161) for further properties of weak derivatives and the spaces L k,p. C 1,2 ([0, a] R k, R) := space of all functions φ : [0, a] R k R which are C 1 in the time variable [0, a] and C 2 in the space variables R k. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.29/95

80 The tame Itô formula Cont d Let X : [ τ, ) Ω R be a pathwise-continuous (not necessarily adapted) R-valued process given by η(0) + t 0 u(s) dw (s) + t 0 v(s) ds, X(t) = t > 0, η(t), τ t 0, (1) where η C := C([ τ, 0], R) and is of bounded variation, u L 2,p loc, p > 4, and v L1,4 loc. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.30/95

81 The tame Itô formula Cont d Let X : [ τ, ) Ω R be a pathwise-continuous (not necessarily adapted) R-valued process given by η(0) + t 0 u(s) dw (s) + t 0 v(s) ds, X(t) = t > 0, η(t), τ t 0, (1) where η C := C([ τ, 0], R) and is of bounded variation, u L 2,p loc, p > 4, and v L1,4 loc. The stochastic integral is a Skorohod integral. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.30/95

82 The tame Itô formula Cont d Set u(t) := 0 for t < 0, and v(t) := η (t), τ t 0, where η is the (usual!) derivative of η. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.31/95

83 The tame Itô formula Cont d Set u(t) := 0 for t < 0, and v(t) := η (t), τ t 0, where η is the (usual!) derivative of η. Let Π : C([ τ, 0], R) R k be the tame projection associated with s 1,, s k [ τ, 0]; that is Π(η) := (η(s 1 ),, η(s k )) for all η C := C([ τ, 0], R). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.31/95

84 The tame Itô formula Cont d For any sample-continuous process X : [ τ, a] Ω R recall its segment X t C([ τ, 0], R), t [0, a]: X t (s) := X(t + s), s [ τ, 0], t [0, a]. Get the tame Itô formula: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.32/95

85 Theorem 2 (Tame Itô Formula) Assume that X is a continuous process defined by (1), where η : [ τ, 0] R is of bounded variation, u L 2,4 loc, and v L 1,4 loc. Suppose φ C1,2 ([0, a] R k, R). Then for all t [0, a] we have a.s. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.33/95

86 Theorem 2 (Tame Itô Formula) Assume that X is a continuous process defined by (1), where η : [ τ, 0] R is of bounded variation, u L 2,4 loc, and v L 1,4 loc. Suppose φ C1,2 ([0, a] R k, R). Then for all t [0, a] we have a.s. φ(t, Π(X t )) φ(0, Π(X 0 )) = + k i= t 0 k i,j=1 t φ x i (s, Π(X s )) dx(s + s i ) t 0 0 φ s (s, Π(X s)) ds 2 φ x i x j (s, Π(X s ))u(s + s i ) si,s j X(s) ds NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.33/95

87 Theorem 2 Cont d where si,s j X(s) := D s+s + i X(s + s j ) + Ds+s i X(s + s j ) and NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.34/95

88 Theorem 2 Cont d where and si,s j X(s) := D + s+s i X(s + s j ) + D s+s i X(s + s j ) D s+s + i X(s + s j ) := lim D s+si X(s + s j + ɛ), ɛ 0+ Ds+s i X(s + s j ) := lim D s+si X(s + s j ɛ). ɛ 0+ NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.34/95

89 Theorem 2 Cont d where and si,s j X(s) := D + s+s i X(s + s j ) + D s+s i X(s + s j ) D s+s + i X(s + s j ) := lim D s+si X(s + s j + ɛ), ɛ 0+ Ds+s i X(s + s j ) := lim D s+si X(s + s j ɛ). ɛ 0+ Proof. Hu, Mohammed and Yan [H.M.Y], Theorem 2.3. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.34/95

90 Corollary 3 Let ψ : R 2 R be of class C 2, and suppose x solves the sdde t η(0) + f ( x(u τ 1 ), x(u) ) du 0 x(t) = t + g ( x(u τ 2 ), x(u) ) dw (u), t > 0 0 η(t), τ < t < 0, τ := τ 1 τ 2, (I) where the coefficients f, g : R 2 R are of class Cb 2, and η C([ τ, 0], R) is of bounded variation. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.35/95

91 Corollary 3 cont d Suppose δ > 0. Then a.s. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.36/95

92 Corollary 3 cont d Suppose δ > 0. Then a.s. dψ ( x(t δ), x(t) ) = ψ x 1 (x(t δ), x(t)) 1 [0,δ) (t) dη(t δ) + ψ x 1 (x(t δ), x(t))1 [δ, ) (t) [ f ( x(t δ τ 1 ), x(t δ) ) dt +g ( x(t δ τ 2 ), x(t δ) ) dw (t δ) ] + ψ x 2 ( x(t δ), x(t) ) f ( x(t τ1 ), x(t) ) dt NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.36/95

93 Corollary 3 cont d + ψ x 2 ( x(t δ), x(t) ) g ( x(t τ2 ), x(t) ) dw (t) + 2 ψ x 1 x 2 ( x(t δ), x(t) ) g ( x(t δ τ2 ), x(t δ) ) ψ x ψ x [δ, ) (t)d t δ x(t) dt ( x(t δ), x(t) ) g ( x(t δ τ2 ), x(t δ) ) 2 1[δ, ) (t) dt ( x(t δ), x(t) ) g ( x(t τ2 ), x(t) ) 2 dt, t > 0. (2) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.37/95

94 Proof of Corollary 3 Suppose t > δ. Apply Theorem 2 with φ := ψ(x 1, x 2 ), X = x, s 1 = δ, s 2 = 0, where x solves the sdde (I). For 0 < t < δ, result follows from classical Itô formula because η is BV. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.38/95

95 Remark In the second term on the right hand side of (2), the (F t ) t 0 -adapted factor ψ x 1 (x(t δ), x(t)) anticipates the differential dw (t δ). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.39/95

96 Lemma 1 Fix a partition point t i in π. Then for a.a. ω Ω, the function [t i, a] C ( [ τ, 0], R ) (t, η) y(t, ω; t i, η) R is tame: That is, there exists a deterministic function F : R + R k R h R l R which is continuous in the time variable R +, of class C 2 b in all space variables Rk, R h, R l, and fixed numbers t 1, t 2,..., t k t, s 1, s 2,..., s h t, µ 1, µ 2,..., µ l [ τ, 0] such that a.s. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.40/95

97 Lemma 1 cont d y(t; t i, η) = F ( t, W (t), W (t 1 ), W (t 2 ),..., W (t k ), s 1, s 2,..., s h, η(µ 1 ), η(µ 2 ),..., η(µ l ) ) for all η C ( [ τ, 0], R ). In particular, for a.a. ω Ω and each t [t i, a], the map C ( [ τ, 0], R ) η y(t, ω; t i, η) R is C 1 (in the Fréchet sense), and NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.41/95

98 Lemma 1 cont d Dy(t, ω; t i, η)(ξ) = l m F ( t, W (t, ω), W (t 1, ω),..., m=1 W (t k, ω), s 1,..., s h, η(µ 1 ),, η(µ m ),..., η(µ l ) ) ξ(µ m ) for all η, ξ C ( [ τ, 0], R ). m F is the partial derivative of F with respect to the variable η(µ m ). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.42/95

99 Lemma 1 cont d Dy(t, ω; t i, η)(ξ) = l m F ( t, W (t, ω), W (t 1, ω),..., m=1 W (t k, ω), s 1,..., s h, η(µ 1 ),, η(µ m ),..., η(µ l ) ) ξ(µ m ) for all η, ξ C ( [ τ, 0], R ). m F is the partial derivative of F with respect to the variable η(µ m ). Proof of Lemma 1: By induction, forward steps along partition intervals [0, t 1 ], (t 1, t 2 ],. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.42/95

100 Warning! NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.43/95

101 Warning! Lemma 1 is false if the Euler approximation y is replaced by the exact solution x of the sdde (I): NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.43/95

102 Warning! Lemma 1 is false if the Euler approximation y is replaced by the exact solution x of the sdde (I): x(t, ω; 0, η) is highly irregular in η! NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.43/95

103 Warning! Lemma 1 is false if the Euler approximation y is replaced by the exact solution x of the sdde (I): x(t, ω; 0, η) is highly irregular in η! This dictates telescoping argument is wrt Euler approximation y and not the solution x of (I) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.43/95

104 Lemma 2 Assume that f, g are C 2 b and let π := {t 0, t 1,, t n } be a partition of [0, a]. For each 1 i n, define the process Λ i : [ τ, 0] Ω R by Λ i : = x ti ( ; ti 1, x ti 1 ( ; 0, η) ) Denote y ti ( ; ti 1, x ti 1 ( ; 0, η) ). x(u) := x(u; 0, η), y(u) := y(u; 0, η) for u [ τ, a]. Then NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.44/95

105 Lemma 2 cont d Λ i (s) = + := (ti +s) t i 1 t i 1 [ f ( x(u τ1 ), x(u) ) (ti +s) t i 1 f ( x ( u τ 1 ), x ( u )] du t i 1 [ g ( x(u τ2 ), x(u) ) 10 j=1 g ( x ( u τ 2 ), x ( u ))] dw (u) Λ i j(s), s [ τ, 0], where NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.45/95

106 Lemma 2 cont d Λ i 1(s) : = + + (ti +s) t i 1 t i 1 (ti +s) t i 1 t i 1 (ti +s) t i 1 t i 1 u u f ( x(v τ1 ), x(v) ) x 1 f ( x(v 2τ 1 ), x(v τ 1 ) ) 1 [τ1, )(v) dv du u f ( x(v τ1 ), x(v) ) u x 1 g ( x(v τ 1 τ 2 ), x(v τ 1 ) ) 1 [τ1, )(v) dw (v τ 1 ) du u u f ( x(v τ1 ), x(v) ) x 1 1 [0,τ1 )(v) dη(v τ 1 ) du NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.46/95

107 Lemma 2 cont d Λ i 2 (s) : = (ti +s) t i 1 + t i 1 (ti +s) t i 1 t i 1 u u f ( x(v τ1 ), x(v) ) x 2 f ( x(v τ 1 ), x(v) ) dv du u u f ( x(v τ1 ), x(v) ) x 2 g ( x(v τ 2 ), x(v) ) dw (v) du NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.47/95

108 Lemma 2 cont d u (ti Λ i +s) t 3 (s) : = i 1 2 f ( x(v τ1 ), x(v) ) t i 1 u x 1 x 2 g ( x(v τ 1 τ 2 ), x(v τ 1 ) ) 1 [τ1, )(v)d v τ1 x(v) dv du NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.48/95

109 Lemma 2 cont d Λ i 4 (s) : = 1 2 (ti +s) t i 1 t i 1 u 2 f ( x(v u x 2 τ1 ), x(v) ) 1 g ( x(v τ 1 τ 2 ), x(v τ 1 ) ) 2 1 [τ1, )(v) dv du NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.49/95

110 Lemma 2 cont d Λ i 4 (s) : = 1 2 Λ i 5(s) : = 1 2 (ti +s) t i 1 t i 1 u 2 f ( x(v u x 2 τ1 ), x(v) ) 1 g ( x(v τ 1 τ 2 ), x(v τ 1 ) ) 2 1 [τ1, )(v) dv du (ti +s) t i 1 t i 1 u u 2 f ( x(v x 2 τ1 ), x(v) ) 2 g ( x(v τ 2 ), x(v) ) 2 dv du NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.49/95

111 Lemma 2 cont d u (ti Λ i +s) t 6 (s) : = i 1 g ( x(v τ2 ), x(v) ) t i 1 u x 1 f ( x(v τ 1 τ 2 ), x(v τ 2 ) ) + (ti +s) t i 1 t i 1 1 [τ2, )(v) dv dw (u) u u g ( x(v τ2 ), x(v) ) x 1 1 [0,τ2 )(v) dη(v τ 2 ) dw (u) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.50/95

112 Lemma 2 cont d Λ i 7 (s) : = (ti +s) t i 1 t i 1 u u g ( x(v τ2 ), x(v) ) x 1 g ( x(v 2τ 2 ), x(v τ 2 ) ) 1 [τ2, )(v) dw (v τ 2 ) dw (u) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.51/95

113 Lemma 2 cont d Λ i 7 (s) : = (ti +s) t i 1 t i 1 u u g ( x(v τ2 ), x(v) ) x 1 g ( x(v 2τ 2 ), x(v τ 2 ) ) 1 [τ2, )(v) dw (v τ 2 ) dw (u) Λ i 8(s) : = (ti +s) t i 1 t i 1 u u 2 g ( x(v τ2 ), x(v) ) x 1 x 2 g ( x(v 2τ 2 ), x(v τ 2 ) ) 1 [τ2, )(v)d v τ2 x(v) dv dw (u) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.51/95

114 Lemma 2 cont d Λ i 9 (s) : = 1 2 (ti +s) t i 1 t i 1 u u τ 1 2 g x 2 1 ( x(v τ2 ), x(v) ) g ( x(v 2τ 2 ), x(v τ 2 ) ) 2 1 [τ2, )(v) dv dw (u) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.52/95

115 Lemma 2 cont d Λ i 9 (s) : = 1 2 (ti +s) t i 1 t i 1 u u τ 1 2 g x 2 1 ( x(v τ2 ), x(v) ) g ( x(v 2τ 2 ), x(v τ 2 ) ) 2 1 [τ2, )(v) dv dw (u) Λ i 10(s) : = 1 2 for all s [ τ, 0]. (ti +s) t i 1 t i 1 u u 2 g ( x(v x 2 τ2 ), x(v) ) 2 g ( x(v τ 2 ), x(v) ) 2 dv dw (u) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.52/95

116 Convention From now on all positive constants will be denoted by the same letter C e.g. C = 2C = 1 2 C = etc.. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.53/95

117 Lemma 3 Suppose f, g Cb 2. Then for any p 1 there is a positive constant C := C(p, a, f, g) such that sup E D u y(t; σ, η) 2p σ τ u,t a < C ( 1 + E η 2p C + sup σ τ s σ E D s η 2p ) ; NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.54/95

118 Lemma 3 Suppose f, g Cb 2. Then for any p 1 there is a positive constant C := C(p, a, f, g) such that sup E D u y(t; σ, η) 2p σ τ u,t a sup σ τ u,t a ξ 1 < C ( 1 + E η 2p C + E D u Dy(t; σ, η)(ξ) 2p < C ( 1 + E η 4p C + sup σ τ s σ sup σ τ s σ E D s η 2p ) ; E D s η 4p ) 1/2 for all η L 4p( Ω, C ( [ τ, 0], R ) ; F σ ) with finite RHS. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.54/95

119 Lemma 4 Suppose f, g Cb 3. Then for any p 1, σ [0, a], sup E D w D u y(t; σ, η) 2p σ τ u,w,t a < C ( 1 + E η 4p C + + sup σ τ s 1,s 2 σ sup σ τ s σ E D s η 4p ) E D s1 D s2 η 4p for all η L 4p( Ω, C ( [ τ, 0], R ) ; F σ ) with RHS finite. C := C(p, a, f, g) > 0 independent of t [σ τ, a], σ, η. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.55/95

120 Lemma 4 Suppose f, g Cb 3. Then for any p 1, σ [0, a], sup E D w D u y(t; σ, η) 2p σ τ u,w,t a < C ( 1 + E η 4p C + + sup σ τ s 1,s 2 σ sup σ τ s σ E D s η 4p ) E D s1 D s2 η 4p for all η L 4p( Ω, C ( [ τ, 0], R ) ; F σ ) with RHS finite. C := C(p, a, f, g) > 0 independent of t [σ τ, a], σ, η. Similar estimate for E D w D u Dy(t; σ, η)(ξ) 2p. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.55/95

121 Proof of Theorem 1 Fix t [σ, a]. Let π := {0 = t 0, t 1, t 2,, t n = a} be a partition of [0, a]. W.l.o.g, assume that σ = 0, t = t n. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.56/95

122 Proof of Theorem 1 Fix t [σ, a]. Let π := {0 = t 0, t 1, t 2,, t n = a} be a partition of [0, a]. W.l.o.g, assume that σ = 0, t = t n. By telescoping and the Markov property for the segments x t and y t ([Mo.1], [Mo.2]), write: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.56/95

123 Proof of Theorem 1 Fix t [σ, a]. Let π := {0 = t 0, t 1, t 2,, t n = a} be a partition of [0, a]. W.l.o.g, assume that σ = 0, t = t n. By telescoping and the Markov property for the segments x t and y t ([Mo.1], [Mo.2]), write: Eφ ( x(t n ; 0, η) ) Eφ ( y(t n ; 0, η) ) = Eφ ( y(t n ; t n, x tn ( ; 0, η)) ) Eφ ( y(t n ; 0, η) ) n { ( = Eφ y(tn ; t i, x ti ( ; t i 1, x ti 1 ( ; 0, η)))) i=1 Eφ ( y(t n ; t i, y ti ( ; t i 1, x ti 1 ( ; 0, η))) )} NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.56/95

124 Proof of Theorem 1 cont d = n i=1 E 1 0 D(φ y) ( t n ; t i, λx ti ( ; t i 1, x ti 1 ( ; 0, η)) + (1 λ)y ti ( ; ti 1, x ti 1 ( ; 0, η)) ) dλ [x ti ( ; ti 1, x ti 1 ( ; 0, η) ) y ti ( ; ti 1, x ti 1 ( ; 0, η) )]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.57/95

125 Proof of Theorem 1 cont d For simplicity, denote each random measure { ( ( D(φ y) tn ; t i, λx ti ; ti 1, x ti 1 ( ; 0, η)) +(1 λ)y ti ( ; t i 1, x ti 1 ( ; 0, η) ))} (ds) by for each λ [0, 1]. D(φ y) i (λ, ds) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.58/95

126 Proof of Theorem 1 cont d Thus, by Fubini s theorem, E ( φ ( x(t n ; 0, η) ) Eφ ( y(t n ; 0, η) ) = 10 j=1 n i= τ E D(φ y) i (λ, ds)λ i j(s) dλ. (3) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.59/95

127 Proof of Theorem 1 cont d Thus, by Fubini s theorem, E ( φ ( x(t n ; 0, η) ) Eφ ( y(t n ; 0, η) ) = 10 j=1 n i= τ E D(φ y) i (λ, ds)λ i j(s) dλ. Estimate each of the 10 terms n 1 0 E { D(φ y) i (λ, ds)λ i j(s) } dλ, j = 1, 2,..., 10 i=1 0 τ on RHS of (3), for any fixed λ [0, 1]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.59/95 (3)

128 Proof of Theorem 1 cont d Let j = 10. Fix λ [0, 1]. Since the Skorohod integral is the adjoint of the Malliavin derivative, a computation via Lemma 2 (Dy tame) gives: I i 10 := = 0 τ 0 τ = Y i 1 + Y i 2, E D(φ y) i (λ, ds)λ i 10(s) E Dφ ( y ( t n ; t i, λx ti + (1 λ)y ti ) Dy ( t n ; t i, λx ti + (1 λ)y ti ) (ds)λ i 10 (s) (4) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.60/95

129 Proof of Theorem 1 cont d Y i 1 := ti t i 1 EX(u)Dy(t n ; t i, λx ti + (1 λ)y ti )(ξ u ) du (5) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.61/95

130 Proof of Theorem 1 cont d Y i 1 := where ti t i 1 EX(u)Dy(t n ; t i, λx ti + (1 λ)y ti )(ξ u ) du X(u) := 1 2 D udφ ( y ( t n ; t i, λx ti + (1 λ)y ti ) (5) u u 2 g ( x(v x 2 τ2 ), x(v) ) g ( x(v τ 2 ), x(v) ) 2 dv, 2 NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.61/95

131 Proof of Theorem 1 cont d Y i 1 := where ti t i 1 EX(u)Dy(t n ; t i, λx ti + (1 λ)y ti )(ξ u ) du X(u) := 1 2 D udφ ( y ( t n ; t i, λx ti + (1 λ)y ti ) (5) u u 2 g ( x(v x 2 τ2 ), x(v) ) g ( x(v τ 2 ), x(v) ) 2 dv, 2 and ξ u L ([ τ, 0], R) is given by ξ u (s) := 1 [ti 1,(t i +s) t i 1 )(u), s [ τ, 0], u [0, a]; NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.61/95

132 Proof of Theorem 1 cont d NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.62/95

133 Proof of Theorem 1 cont d Y i 2 := where ti t i 1 EZ(u)D u Dy(t n ; t i, λx ti + (1 λ)y ti )(ξ u ) du Z(u) := 1 2 Dφ( y ( t n ; t i, λx ti + (1 λ)y ti ) (5) u u 2 g ( x(v x 2 τ2 ), x(v) ) g ( x(v τ 2 ), x(v) ) 2 dv. 2 NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.62/95

134 Proof of Theorem 1 cont d By the linearization of (II) and Gronwall s lemma, we get sup ξ L ([ τ,0],r) ξ 1 sup σ τ t a E Dy(t; σ, η)(ξ) 2p C (6) for every p 1. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.63/95

135 Proof of Theorem 1 cont d Using (similar) moment estimates on the solution, the Euler approximation and its Fréchet and Malliavin derivatives: Y i 1 C(1 + E η 3 C)(t i t i 1 ) 2 (7) Y i 2 C(1 + E η 4 C)(t i t i 1 ) 2 (8) Positive constants C are independent of η and the partition {t 1, t 2,, t n }. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.64/95

136 Proof of Theorem 1 cont d Putting things together: n n 1 I10 i = i=1 i=1 0 0 τ E { D(φ y) i (λ, ds)λ i 10(s) } dλ n C(1 + E η 3 C ) (t i t i 1 ) 2 i=1 + C(1 + E η 4 C) C(1 + E η 4 C) π. n (t i t i 1 ) 2 i=1 (9) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.65/95

137 Proof of Theorem 1 cont d Develop estimates similar to (9) for the 9 cases j = 1, 2, 3, 4, 5, 6, 7, 8, 9. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.66/95

138 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.67/95

139 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. Quasitame memory dependence. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.67/95

140 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. Quasitame memory dependence. Mixed history dependence: discrete and quasitame. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.67/95

141 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. Quasitame memory dependence. Mixed history dependence: discrete and quasitame. Multidimensional Brownian motion. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.67/95

142 REFERENCES [A] Ahmed, T. A. Stochastic Functional Differential Equations with Discontinuous Initial Data, M.Sc. Thesis, University of Khartoum, Sudan (1983).(< ) [AHMP] Arriojas, M., Hu, Y., Mohammed, S.-E.A. and Pap, G., A Delayed Black and Scholes Formula, Stochastic Analysis and Applications, Vol. 25, 2, (2007), ( ) [B.B] Baker, C. T. H. and Buckwar, E., Numerical analysis of explicit one-step methods for stochastic delay differential equations., LMS J. Comput. Math. 3 (2000), (electronic).( ) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.68/95

143 REFERENCES cont d [B.T] [B.M] [B.S] Bally, V., and Talay, D., The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields, 104 (1996), (< ) Bell, D. and Mohammed, S.-E. A., The Malliavin calculus and stochastic delay equations, Journal of Functional Analysis 99 No. 1 (1991), Buckwar, E. and Shardlow, T., Weak approximation of stochastic differential delay equations, IMA J. Numer. Anal. 25 (2005), (< ) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.69/95

144 REFERENCES cont d [B.V] Beuter, A. and Vasilakos, K., Effects of noise on a delayed visual feedback system. J. Theor. Biology 165 (1993), (< ) [BGMTS] Buldú, J.M., Garcia-Ojalvo, J., Mirasso, C.R., Torrent, M.C., and Sancho, J.M., Effect of external noise correlation in optical coherence resonance. Phys. Rev. E 64 (2001), (< ) [CK-HL] E. Clément, A. Kohatsu-Higa and D. Lamberton, A duality approach for weak approximation of stochastic differential equations. (< ) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.70/95

145 REFERENCES cont d [GRP] [Hu] Goldobin, D., Rosenblum, M., and Pikovsky, A., Coherence of noisy oscillators with delayed feedback Physica A 327 (2003), (< ) Hu, Y., Strong and weak order of time discretization schemes of stochastic differential equations, In Séminaire de Probabilités XXX, ed. by J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1626, Springer-Verlag, 1996, [H.M.Y] Hu, Y., Mohammed, S.-E. A., and Yan, F., Discrete-time approximations of stochastic delay equations: The Milstein scheme, The Annals of Probability, Vol. 32, No. 1A, (2004), (< ) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.71/95

146 REFERENCES cont d [K] Kohatsu-Higa, Y., Weak approximations. A Malliavin calculus approach. Math. Comput. 70(233) (2001), (< ) [LMBM] Longtin, A., Milton, J.G., Bos, J. and Mackey, M.C., Noise and critical behavior of the pupil light reflex at oscillation onset. Phys. Rev. A 41 (1990), (< ) [K.P] Kloeden, P. E. and Platen E., Numerical Solutions of Stochastic Differential Equations, Springer-Verlag, New York-Berlin (1995).(< ) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.72/95

147 REFERENCES cont d [Kü.P] [Ma.1] [Ma.2] Küchler, U. and Platen, E. Strong Discrete Time Approximation of Stochastic Differential Equations with Time Delay. Mathematics and Computer Simulation 54 (2000), (< ) Masoller, C., Numerical investigation of noise-induced resonance in a semiconductor laser with optical feedback., Physica D (2002), Masoller, C., Distribution of residence times of timedelayed bistable systems driven by noise. Phys. Rev. Lett. 90(2) (2003), (< ) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.73/95

148 REFERENCES cont d [Mo.1] [Mo.2] [N.P] Mohammed, S.-E. A., Stochastic Functional Differential Equations, Research Notes in Mathematics, no. 99, Pitman Advanced Publishing Program, Boston-London-Melbourne (1984).(< ) Mohammed, S.-E. A., Stochastic differential systems with memory: Theory, examples and applications. In Stochastic Analysis", Decreusefond L. Gjerde J., Øksendal B., Ustunel A.S., edit., Progress in Probability 42, Birkhauser (1998), 1-77.(< ) Nualart, D. and Pardoux, E., Stochastic calculus with anticipating integrands, Probability Theory and Related fields 78 (1988), NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.74/95

149 REFERENCES cont d [Nu.1] [Nu.2] [Y] Nualart, D., The Malliavin Calculus and Related Topics, Springer-Verlag, 1995.(< ) Nualart, D., Analysis on Wiener Space and Anticipating Stochastic Calculus, Ecole d Été de Probabilités, Saint Flour Yan, F., Topics on Stochastic Delay Equations, Ph.D. Dissertation, Southern Illinois University at Carbondale, August, NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.75/95

150 THE END NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.76/95

151 THANK YOU! NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.77/95

152 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.78/95

153 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. Quasitame memory dependence. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.78/95

154 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. Quasitame memory dependence. Mixed history dependence: discrete and quasitame. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.78/95

155 Extensions Extensions cover: Nonlinear R d -valued sdde s with several delays. Quasitame memory dependence. Mixed history dependence: discrete and quasitame. Multidimensional Brownian motion. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.78/95

156 Extensions Notation Let W (t) := (W 1 (t), W 2 (t),, W m (t)) t 0, be m-dimensional standard Brownian motion on a filtered probability space (Ω, F, (F t ) t 0, P ). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.79/95

157 Extensions Notation Let W (t) := (W 1 (t), W 2 (t),, W m (t)) t 0, be m-dimensional standard Brownian motion on a filtered probability space (Ω, F, (F t ) t 0, P ). Consider a finite number of delays {τ1 i : 1 i k 1 }, {τ j,l 2 : 1 j k 2,l, 1 l m}, with maximum delay τ := max{τ1, i τ j,l 2 : 1 i k 1, 1 j k 2,l, 1 l m}. We will designate the memory in our sfde by a collection of tame projections NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.79/95

158 Extensions -cont d Π 1 : C := C([ τ, R d ) R d 1, Π 2,l : C R d 2,l Π 1 (η) := (η(τ 1 1 ), η(τ 2 1 ),, η(τ k 1 1 )), Π 2,l (η) := (η(τ 1,l 2,l 2 ), η(τ2 ),, η(τ k 2,l,l 2 )) for all η C, and quasitame projections Π 1 q : C R dq 1, Π 2,l q : C R dq 2,l where d 1 = k 1 d, d q 1 = k 2d, d 2,l = k 2,l d, d q 2,l = k 2,l d are integer multiples of d, for 1 l m. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.80/95

159 Extensions -cont d The quasitame projections are of the form: Π 1 q(η) := ( 0 τ σ 1 1(η(s))µ 1 1(s) ds,, 0 τ 0 τ σ 1 2(η(s))µ 2 2(s) ds, σ 1 k 2 (η(s))µ 1 k 2 (s) ds) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.81/95

160 Extensions cont d Π 2,l q (η) := ( 0 τ σ 2 1 (η(s))µ2 1 (s) ds, 0, 0 τ τ σ2 2 (η(s))µ2 2 (s) ds, σ 2 k 2,l (η(s))µ 2 k 2,l (s) ds) for all η C. The functions σ 1 i, σ2 j, µ1 i, µ2 j are smooth. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.82/95

161 Extensions cont d Let f : R + R d 1 R dq 1 R d, g l : R + R d 2,l R dq 2,l R d be functions of class C 1 in the first variable and Cb 3 space variables. in all NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.83/95

162 Extensions cont d Consider the sfde dx(t) = f(t, Π 1 (x t ), Π 1 q(x t )) dt m + g l (t, Π 2,l (x t ), Π 2,l q (x t )) dw l (t), σ < t < a, l=1 with initial path x σ = η H 1, ([ τ, 0], R d ). (III) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.84/95

163 Extensions -cont d Let π := {t m,, t 0, t 1, t 2,, t n } be a partition of [ τ, a] with mesh π. The Euler approximations y of x are given by dy(t) = f( t, Π 1 (y t ), Π 1 q(y t )) dt m + g l ( t, Π 2,l (y t ), Π 2,l q (y t )) dw l (t), l=1 with initial path y σ = η H 1, ([ τ, 0], R d ). σ < t < a, (IV) NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.85/95

164 Extensions cont d Under sufficient regularity hypotheses on the coefficients of (III), one gets weak convergence of order 1 for the Euler approximations y in (IV) to the exact solution x. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.86/95

165 Theorem 4 Let φ : R R be of class Cb 3. In the sfde (III), let f, g l, 1 l m, be C 1 in the time variable and Cb 3 in all space variables. Let x( ; σ, η) be the unique solution of (III) with initial process η H 1, ( [ τ, 0], R d). NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.87/95

166 Theorem 4 Let φ : R R be of class Cb 3. In the sfde (III), let f, g l, 1 l m, be C 1 in the time variable and Cb 3 in all space variables. Let x( ; σ, η) be the unique solution of (III) with initial process η H 1, ( [ τ, 0], R d). Denote by y( ; σ, η) the Euler approximation to x( ; σ, η) defined by (IV). Let η π be the piecewise-linear approximation of η. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.87/95

167 Theorem 4 Let φ : R R be of class Cb 3. In the sfde (III), let f, g l, 1 l m, be C 1 in the time variable and Cb 3 in all space variables. Let x( ; σ, η) be the unique solution of (III) with initial process η H 1, ( [ τ, 0], R d). Denote by y( ; σ, η) the Euler approximation to x( ; σ, η) defined by (IV). Let η π be the piecewise-linear approximation of η. Then there is a positive constant C and a positive integer q such that Eφ(x(t; σ, η)) Eφ(y(t; σ, η π )) C(1 + E η q 1, ) π for all η H 1, ( [ τ, 0], R d), all t [σ τ, a], and all σ [0, a]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.87/95

168 Theorem 4 cont d The constant C may depend on a, q and the test function φ, but is independent of π, η, t [σ τ, a] and σ [0, a]. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.88/95

169 Theorem 4 cont d The constant C may depend on a, q and the test function φ, but is independent of π, η, t [σ τ, a] and σ [0, a]. Proof: Very similar to that of Theorem 1: Main difference is a straightforward application of the classical Itô formula combined with the tame Itô formula. NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.88/95

170 Lemma 5 Let ψ : R + R 3 R be of class C 1 in the time-variable and C 2 b in the three space variables x 1, x 2, x 3. Suppose x solves the sfde (III) (for d = 1) with coefficients satisfying the hypotheses of Theorem 4. Assume that h, µ are smooth functions. Let δ > 0. Then: NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.89/95

171 Lemma 5 Let ψ : R + R 3 R be of class C 1 in the time-variable and Cb 2 in the three space variables x 1, x 2, x 3. Suppose x solves the sfde (III) (for d = 1) with coefficients satisfying the hypotheses of Theorem 4. Assume that h, µ are smooth functions. Let δ > 0. Then: dψ ( t, x(t δ), x(t), = ψ t 0 (t, x(t δ), x(t), h(x(t + s))µ(s) ds ) δ 0 δ h(x(t + s))µ(s) ds) dt NUMERICSOF STOCHASTIC SYSTEMSWITH MEMORY p.89/95