Qualitative behaviour of numerical methods for SDEs and application to homogenization

Transcription

1 Qualitative behaviour of numerical methods for SDEs and application to homogenization K. C. Zygalakis Oxford Centre For Collaborative Applied Mathematics, University of Oxford. Center for Nonlinear Analysis, Carnegie Mellon University, 20/10/2011 K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 1 / 45

2 Outline 1 Modified Equations ODE theory. Main idea for SDEs. Different numerical methods and Associated Modified Equations. Numerical examples. 2 Application to Homogenization Long time behaviour and homogenization. Numerical algorithms/results. From homogenization to averaging in cellular flows. 3 Higher order numerical methods based on modified equations. Key idea. One simple example. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 2 / 45

3 Motivating example Introduction K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 3 / 45

4 Introduction Interesting Question The two numerical methods have the same order of convergence but completely different qualitative behaviour. Is there a way to distinguish between these two methods? A very powerful tool for addressing this question is backward error analysis (modified equations). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 4 / 45

5 Introduction Modified equations for ODEs Ordinary Differential Equations dx dt = f (x), and let x n be a numerical approximation of x of order p: x(nh) x n = O(h p ). Can I find X (t) satisfying another ODE (modified equation) such that: X (nh) x n = O(h p+q ). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 5 / 45

6 Introduction Euler method-one dimension Ordinary Differential Equations x n+1 = x n + hf (x n ). Modified equation: since dx dt = f (X ) h 2 f (X )f (X ), X (nh) x n = O(h 2 ). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 6 / 45

7 Introduction Ordinary Differential Equations Sketch proof dx dt = f (X ) + hg(x ). X (h) = X (0) + h 0 (f (X (s)) + hg(x (s))) ds = X (0) + hf (X (0)) + h 2 g(x (0)) + h2 2 f (X (0))f (X (0)) + O(h 3 ). Assume x 0 = X (0) then X (h) x 1 = h (g(x 2 (0)) + 1 ) 2 f (X (0))f (X (0)) + O(h 3 ), and thus g(x) = 1 2 f (x)f (x). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 7 / 45

8 Introduction Stochastic Differential Equations Stochastic Differential Equations and Numerical Methods dx = u(x)dt + σ(x)dw t, Euler method: θ-milstein method: x n+1 = x n + hu(x n ) + hσ(x n )ξ n, x n+1 = x n +θhu(x n+1 )+(1 θ)hu(x n )+ hσ(x n )ξ n + h 2 σ(x n)σ (1) (x n )(ξ 2 n 1), where ξ n N (0, 1). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 8 / 45

9 Introduction Stochastic Differential Equations Weak and Strong Convergence Weak convergence: We look at E(φ(x(nh))) E(φ(x n )). Strong convergence: We look at E x(nh) x n. In general the weak and strong order of convergence of a numerical method NEEDS NOT to be the same!!! K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 9 / 45

10 Introduction Statement of the Problem Stochastic Differential Equations Let x(t) satisfy the following SDE: dx = u 1 (x)dt + σ 1 (x)dw t, and x n be its numerical approximation at T = nh by a weak p-order method i.e E(φ(x(T ))) E(φ(x n )) = O(h p ), φ C. We want to develop a procedure that allows us to evaluate the properties of our weak numerical scheme. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 10 / 45

11 Introduction Statement of the Problem Stochastic Differential Equations Let x(t) satisfy the following SDE: dx = u 1 (x)dt + σ 1 (x)dw t, and x n be its numerical approximation at T = nh by a weak p-order method i.e E(φ(x(T ))) E(φ(x n )) = O(h p ), φ C. We want to develop a procedure that allows us to evaluate the properties of our weak numerical scheme. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 10 / 45

12 First Modified Equation Introduction Stochastic Differential Equations We want to find a modified SDE of the form (i.e., find v 2 and σ 2 ) d x = [u 1 ( x) + hu 2 ( x)] + [σ 1 ( x) + hσ 2 ( x)] dw t, for which E(φ( x(t ))) E(φ(x n )) = O(h p+1 ), φ C. For the rest of the talk we concentrate in the case where p = 1. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 11 / 45

13 Main Idea Generators for ODEs and SDEs ODE: dx = h(x)dt, Lu := h(x) x u. SDE: dx = h(x)dt + σ(x)dw t, Lu := h(x) x u σ(x)σt (x) : x x u. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 12 / 45

14 Main Idea Backward Kolmogorov Equation u t = Lu, u(x, 0) = φ(x). Then u(x, t) = E(φ(x(t)) x(0) = x). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 13 / 45

15 Stochastic B-series Main Idea By integrating over time the backward Kolmogorov Equation and taking a Taylor expansion of u(x, s) around s = 0, we obtain, (assuming appropriate smoothness of the drift and diffusion term) u(x, h) φ(x) = k=0 h k+1 (k + 1)! Lk+1 φ(x). Note that in the case where φ(x) = x, σ(x) = 0, this expansion correspond to the B-series expansion of the ODE dx = v 1 (x)dt. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 14 / 45

16 Main Idea Local Error/Global Error A weak first order numerical method has the following expansion u num (x, h) φ(x) = hlφ(x) + h 2 L e φ(x) + O(h 3 ), and so ( ) 1 u(x, h) u num (x, h) = h 2 2 L2 φ(x) L e φ(x), Local Error which implies that u(x, T ) u num (x, T ) = O(h). Global Error K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 15 / 45

17 Main Idea Generator of the Modified Equation Remember that the 1-st modified equation is of the form d x = [u 1 ( x) + hu 2 ( x)] + [σ 1 ( x) + hσ 2 ( x)] dw t. Its generator L can be written as L = L 0 + hl 1 + h 2 L 2, where L 0 is the generator of the original SDE and L 1 φ := u 2 (x) dφ dx + σ 1(x)σ 2 (x) d 2 φ dx 2. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 16 / 45

18 Main Equation Main Idea If we now subtract the Taylor expansion of the numerical method from the stochastic B-series of the modified equation we see that in order for the local error to be O( t 3 ) we need L 1 φ = L e φ 1 2 L2 0φ, φ C. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 17 / 45

19 Different Numerical Methods Euler-Maryama Method In the case of Euler-Maryama method in the case of multiplicative noise it turns out that a modified equation does not exist since L 1 φ + σ3 1 (x) σ (1) 1 2 (x)φ(3) (x). as L 1 is a second order partial differential operator!!! K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 18 / 45

20 θ-milstein Method Different Numerical Methods u 2 (x) = σ 2 (x) = ( θ 1 ) ( v 1 (x)v (1) 2 1 (x) + σ2 1 (x) 2 ( θ 1 ) σ 1 (x)v (1) 1 2 (x) 1 2 v 1(x)σ (1) v (2) 1 (x) ), 1 (x) σ2 1 (x) 4 σ (2) 1 (x). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 19 / 45

21 Numerical examples Geometric Brownian motion SDEs Driven by Multiplicative Noise dx = µxdt + σxdw t, d X = [(µ h2 ) ] µ2 X dt + σ X (1 hµ) dw t error error original SDE modified SDE 10 3 original SDE modified SDE t First moment t Second moment K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 20 / 45

22 Numerical examples Modified equations Linear SDEs with additive noise dx = Axdt + ΣdW t, Numerical Approximation: x(h) = A(h)x + f (h, ω). Example (Euler-Maryama): A(h) = (I + ha), f (h, ω) = Σ hξ. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 21 / 45

23 Numerical examples Modified equations Modified Equation and its coefficients dx = Ãxdt + ΣdW t, Ã = log(a(h)), h eãh Σ Σ T eãt h Σ Σ T = ÃJ + JÃ T, where J = E(ff T ). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 22 / 45

24 Numerical examples Orstein Uhlenbeck Process Modified equations Forward Euler: Backward Euler: dx = γxdt + σdw t. log(1 γh) Ã =, h 2 log(1 γh) Σ = σ (1 γh) 2 1. log(1 + γh) Ã =, h 2 log(1 + γh) Σ = σ 1 (1 + γh) 2. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 23 / 45

25 Invariant Measure Numerical examples Modified equations lim t E(x2 (t)) = lim t E(x2 (t)) = σ 2 2γ γ 2, Forward Euler h σ 2 (1 + γh) 2γ + γ 2, Backward Euler. h numerical solution modified equation numerical solution modified equation Var(x) 0.58 Var(x) t Forward Euler t Backward Euler Figure: lim t E(x 2 (t)) as a function of h. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 24 / 45

26 An application to homogenization Passive Tracers, Effective Diffusivity Long Time Behaviour and Homogenization dx = v(x)dt + σdw t, where v(x) is a periodic function. It is possible to show using homogenization that E(x(t) x(t)) lim = K. t 2t We will refer to K as the effective diffusivity matrix K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 25 / 45

27 An application to homogenization Long Time Behaviour and Homogenization Velocity Field of Interest Example We are interested in the following 2-dimensional incompressible velocity field v(x) = Ψ(x), where Ψ(x) = sin x 1 sin x 2 Result In this case it is known that K = DI 2, where D R depending on σ for passive tracers and that the following result is true for passive tracers D(σ) σ, σ 1 K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 26 / 45

28 An application to homogenization Key Property of the Velocity Field Long Time Behaviour and Homogenization Our velocity field v(x) can be written as v(x) = ( ) ( 1/2 1/2 sin(x +1/2 1 + x 2 ) + 1/2 = 2 ( d j v j ej, x ), j=1 where e j, d j R 2 with the property e j, d j = 0. ) sin(x 1 x 2 ), This is a key property for the construction of our method which is a stochastic extension of a splitting method proposed by Quispel K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 27 / 45

29 An application to homogenization Numerical algorithm/results Description of the Method: The method in the case of passive tracers involves these 3 steps: ( Step 1: Solve ẋ 1 = d 1 v 1 e1, x 1 ), ( Step 2: Solve ẋ 2 = d 2 v 2 e2, x 2 ), Step 3: Solve ẋ 3 = σ β 1. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 28 / 45

30 An application to homogenization The Deterministic Case Numerical algorithm/results Splitting method for σ = 0. Euler method for σ = 0. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 29 / 45

31 The Case σ 1 An application to homogenization Numerical algorithm/results Splitting method for σ = Euler method for σ = K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 30 / 45

32 An application to homogenization Calculating Effective Diffusivities Numerical algorithm/results Volume preserving Euler!Maruyama 10!1 10! !3 K K 10!1 10!4 10!2 10!5 10!3 10!2 10! Comparison of the two methods.! 10!6 10!3 10!2 10!1 Splitting Method.! K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 31 / 45

33 Mean Hamiltonian An application to homogenization Numerical algorithm/results We apply Itô s formula to H = Ψ we obtain dψ dt = σ2 Ψ + M.T which implies that the mean Hamiltonian decays like e σ2 t K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 32 / 45

34 An application to homogenization Numerical algorithm/results Numerical calculation of the mean Hamiltonian with the two methods Euler Method Stochastic Splitting Method Theory Mean Hamiltonian ! t x 10 4 Figure: Mean value of the Hamiltonian as a function of time, for t = 10 1, σ = K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 33 / 45

35 An application to homogenization Numerical algorithm/results Modified Equations for the Euler Method dx = + σ ( v(x) t 2 ( v(x))v(x) σ2 t 4 ) dw t. ( 1 t 2 v T (x) ) v(x) dt dψ dt = t 2 (cos2 x 1 + cos 2 x 2 )Ψ σ 2 Ψ(1 + t cos x 1 cos x 2 ) + σ2 2 t (cos 2 x 1 cos 2 x 2 Ψ Ψ 3 ) + M t. 4 K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 34 / 45

36 An application to homogenization From Homogenization to averaging in cellular flows. Statement of the problem u(x) = Ψ(x), Ψ(x) = 1 π sin πx 1 sin πx 2. X t satisfies the following SDE dx t = Au(X t )dt + 2dW t. Exit time problem τ + Au τ = 1 in D, τ = 0 on D, where D = [ L/2, L/2] [ L/2, L/2]. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 35 / 45

37 An application to homogenization Known results and open questions From Homogenization to averaging in cellular flows. 1 A fixed, L, homogenization (τ ). 2 L fixed, A, averaging (τ 0). 3 If D = B L a disk of radius L, then τ(x) L 2 x 2. 4 A = L α and L α < 4, homogenization (τ L 2 α/2 ). α > 4, averaging (τ L 2 α/2 ). Question How about α = 4? K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 36 / 45

38 An application to homogenization Numerical investigations I From Homogenization to averaging in cellular flows. 2 Define Y t = X At, then dy t = u(y t )dt + A dw t and τ(y) = Aτ(x) Asymptotic behaviour of exit time (α = 1) Exit problem from a disk (L = 40) K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 37 / 45

39 An application to homogenization Numerical investigations II, Case α = 4 From Homogenization to averaging in cellular flows. Asymptotic behaviour of exit time (α = 4) Comparison of c.d.f K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 38 / 45

40 Higher order methods Key idea i) Choose a numerical method for the original SDE. ii) Write down a suitably chosen SDE different than the original one (this SDE depends on the choice of the method from step i). iii) Apply the numerical method from step i to the SDE from step ii. Example dx t = u 1 (X t)dt + σ 1 (X t) i) x n+1 = x n +θhu 1 (x n+1 )+(1 θ)hu 1 (x n)+ hσ 1 (x n)ξ n + h 2 σ 1(x n)σ (1) 1 (xn)(ξ2 n 1) ii) dx t = ũ(x t)dt + σ(x t), ũ = u 1 hu 2, σ = σ 1 hσ 2 iii) x n+1 = x n + θhũ(x n+1 ) + (1 θ)hũ(x n) + h σ(x n)ξ n + h 2 σ 1(x n)σ (1) 1 (xn)(ξ2 n 1) K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 39 / 45

41 Higher order methods Application to an economy model for asset prices dx 1 = β 1 X 1 X 2 dw 1, dx 2 = (X 2 X 3 )dt + β 2 X 2 dw 2, dx 3 = α(x 2 X 3 )dt, N. Hofmann, E. Platen, M. Schweizer. Option pricing under incompletness and stochastic volatility. Mathematical Finance, 2(3): , (1992). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 40 / 45

42 Higher order methods Numerical Investigations Error for E(X1 2). Nonstiff case α = 1. Error for E(X 1 2 ). Very stiff case α = 100. Error for E(X1 2). Stiff case α = 25. Error for E(X 2 2 ). Stiff case α = 25. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 41 / 45

43 Conclusions Summary 1 It is not always possible to write down a modified Itô SDE for a given numerical method. 2 In the case of linear SDEs with additive noise it is possible to write down an -modified equation that the numerical method satisfy exactly in the weak sense. 3 It is possible to generalize ideas from the backward error analysis of ODEs to SDEs. 4 Modified equations can be used as a tool for constructing higher order methods. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 42 / 45

44 Summary Future work 1 Find modified equations for numerical methods with respect to strong convergence. 2 Give a rigorous explanation for failing to find a modified SDE for the Euler method in case of multiplicative noise. 3 Use modified equations to characterize the invariant measure approximated by different numerical schemes. 4 Compare exit times from a square for different starting points, with the ones of the effective Brownian motion. 5 Study exit times in case where the inertia is important (inertial particles). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 43 / 45

45 Acknowledgements Thank for your attention! Collaborators: A. Abdulle (EPFL), D. Cohen (Basel), G. Iyer (CMU), G. Pavliotis (Imperial), A. M. Stuart (Warwick), G. Villmart (ENS Cachan). Funding: David Crighton Fellowship and award KUK-C , made by King Abdullah University of Science and Technology (KAUST). K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 44 / 45

46 References Ernst Hairer, Christian Lubich, and Gerhard Wanner. Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, (2002). G.R.W. Quispel and D.I. McLaren, Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows, J. Comp. Phys, , (2003). T. Shardlow. Modified equations for stochastic differential equations.bit, 46(1): , (2006). I. Lenane, K. Burrage and G. Lythe. Numerical methods for second order stochastic equations. SIAM J.Sci.Comp, 29(1): , (2008). G. A. Pavliotis, A. M. Stuart, K. C. Zygalakis. Calculating Effective Diffusivities in Vanishing Molecular Diffusion, J. Comp. Phys. 228, (2009). K. C. Zygalakis. On the existence and the applications of modified equations for stochastic differential equations. SIAM J. Sci. Comput.. 33, (2011). A. Abdulle, D. Cohen, G. Villmart, K. C. Zygalakis. High order weak methods for stochastic differential equations based on modified equations. Submitted to SIAM J. Sci. Comput.. G. Iyer, T. Komorowski, A. Novikov, L. Ryzhik. From homogenization to averaging in cellular flows. (Submitted) A. Abdulle, G. Villmart, K. C. Zygalakis. Explicit higher order methods for stiff stochastic differential equations. In preparation. K. C. Zygalakis (University of Oxford) Modified Equations for SDEs 45 / 45