Introduction to numerical simulations for Stochastic ODEs

Transcription

1 Introduction to numerical simulations for Stochastic ODEs Xingye Kan Illinois Institute of Technology Department of Applied Mathematics Chicago, IL August 9, 2010

2 Outline 1 Preliminaries 2 Numerical scheme for solving SODEs Euler-Maruyama scheme Milstein scheme 3 MATLAB simulation

3 Brownian Motion(Wiener Process) Definition[1] B 0 (ω) = 0 a.s.; the sample paths t B t (ω) are a.s. continuous; B t (ω) has stationary independent increments; the increments B t (ω) B s (ω) has the normal distribution with mean 0 and variance t s, i.e. B t (ω) B s (ω) N(0, t s) for any 0 s < t. 2 A Scalar Brownian Motion B t t

4 SODE driven by Brownian motion Definition[1] An equation of the form dx t = a(t, X t )dt + b(t, X t )db t where functions a(t, x), b(t, x) are given, B t is a Brownian motion process and X t is the unknown process is called a stochastic ordinary differential equation(sode) driven by Brownian motion. Integral form t t X t = X t0 + a(s, X s )ds + b(s, X s )db s t 0 t 0

5 Itô formula t t U(t, X t ) = U(t 0, X t0 ) + L 0 U(s, X s )ds + L 1 U(s, X s )db s t 0 t 0 where L 0 = t + a x b2, L 1 = b x 2 x

6 Examples of SODEs Black-Scholes model ds t = µs t dt + σs t db t Logistic equation(perturbed by multiplicative noise) ẋ = rx(1 x) + ɛx B t Harmonic Oscillator(perturbed by additive noise) ẋ = y + ɛ B 1 t ẏ = x + ɛ B 2 t

7 Examples of SODEs(cont d) Simple Pendulum(perturbed by additive noise) ẋ = y + ɛ B 1 t ẏ = sin(x) + ɛ B 2 t Lorenz equations(perturbed by additive noise) ẋ = σ(y x) + ɛ B 1 t ẏ = x(τ z) y + ɛ B 2 t ż = xy βz + ɛ B 3 t

8 Euler-Maruyama scheme for SDE t t X t = X t0 + a(s, X s )ds + b(s, X s )dw s t 0 t 0 Applying Itô formula on the integrands over the time interval [, +1 ] a(s, X s ) =a(, X tn ) + L 0 a(u, X u )du + b(s, X s ) =b(, X tn ) + L 0 b(u, X u )du + t n s L 1 a(u, X u )dw u L 1 b(u, X u )dw u

9 X tn+1 = X tn + + [a(, X tn )+ [b(, X tn )+ = X tn + a(, X tn ) L 0 a(u, X u )du + Euler-Maruyama scheme L 0 b(u, X u )du + ds + b(, X tn ) L 1 a(u, X u )dw u ]ds L 1 b(u, X u )dw u ]dw s dw s +R 1 (+1, ) X tn+1 = X tn + a(, X tn ) ds + b(, X tn ) dw s

10 Milstein scheme for SDE t t X t = X t0 + a(s, X s )ds + b(s, X s )dw s t 0 t 0 Applying Itô formula on a(s, X s ), b(s, X s ) over the time interval [, +1 ] Applying Itô formula again on L 1 b(u, X u ) over the time interval [, +1 ]

11 X tn+1 = X tn [a(, X tn )+ [b(, X tn )+ { u + L 0 a(u, X u )du + [L 1 b(, X tn ) L 0 L 1 b(v, X v )dv + = X tn + a(, X tn ) +L 1 b(, X tn ) L 0 b(u, X u )du]dw s u L 1 a(u, X u )dw u ]ds L 1 L 1 b(v, X v )dw v ]dw u }dw s ds + b(, X tn ) dw u dw s + R 2 (+1, ) dw s

12 Milstein scheme X tn+1 = X tn + a(, X tn ) +L 1 b(, X tn ) = X tn + a(, X tn ) ds + b(, X tn ) +b(, X tn ) b x (, X tn ) dw u dw s ds + b(, X tn ) dw u dw s dw s dw s

13 MATLAB GUI is short for Graphic User Interface Simulate Brownian motion sample paths Simulate Itô SDEs Specify different integration method to be used between Euler-Maruyama and Milstein schemes Set the time interval to be considered Set stepsize

14 Example 1: GUI for Brownian motion sample paths B j = B j 1 + db j B 0 = 0 j = 1, 2,, N N = T /h 2 A Scalar Brownian Motion stepsize = 0.1 final time = 20.0 Scalar Planar 3D B t Quit Plot t

15 Example 2: GUI for sample paths of solutions of SDEs 1 Stochastic differential equation 2 Stochastic differential equations 3 Stochastic differential equations Euler scheme: blue Milstein scheme: red 0.9 dx_t = -x^2 dy_t = 2*x-y^2 dz_t = sin(x) dt + x/100 dt + y/100 dt + z/100 dw_t dw_t dw_t Z(t) stepsize = final time = X_0= 0.1 Y_0= 0.1 Z_0= 0.1 Euler Milstein both Quit Proceed 0.1 Y(t) X(t)

16 Using MATLAB GUI to solve 1D, 2D and 3D SODE Black-Scholes model ds t = µs t dt + σs t db t with S 0 = 40, µ = 0.08, σ = numerical solution using Euler scheme X(t) t

17 Logistic equation(perturbed by multiplicative noise) with x 0 = 10, r = 0.4, ɛ = 0.1 ẋ = rx(1 x)dt + ɛxdb t 10 numerical solution using Milstein scheme X(t) t

18 Harmonic Oscillator(perturbed by additive noise) with x 0 = 1, y 0 = 0.1, ɛ = 0.1 ẋ = y + ɛ B 1 t ẏ = x + ɛ B 2 t 1.5 numerical solution using Euler scheme Y(t) X(t)

19 Simple pendulum(perturbed by additive noise) with x 0 = 0.1, y 0 = 2, ɛ = 0.1 ẋ = y + ɛ B 1 t ẏ = sin(x) + ɛ B 2 t 2 numerical solution using Milstein scheme Y(t) X(t)

20 Lorenz equations(perturbed by additive noise) ẋ = σ(y x) + ɛ B 1 t ẏ = x(τ z) y + ɛ B 2 t ż = xy βz + ɛ B 3 t with x 0 = 0.1, y 0 = 2, z 0 = 1.2, σ = 0.2, τ = 0.6, β = 0.7, ɛ = 0.1 Euler scheme: blue Milstein scheme: red Z(t) Y(t) X(t)

21 Fima C Klebaner Introduction to Stochastic Calculus with Applications 2nd Ed. Imperial College Press, London, 2005.