The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method

Yousef Alnafisah

doi:10.1155/2018/3805042

Abstract

Multiple stochastic integrals of higher multiplicity cannot always be expressed in terms of simpler stochastic integrals, especially when the Wiener process is multidimensional. In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.

Highlights

Numerical analysis for stochastic differential equation (SDE) has seen a considerable development in recent years
In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program
Kloeden and Platen [1] described a method based on the stochastic Taylor series expansion but the major difficulty with this approach is that the double stochastic integrals cannot be expressed in terms of simpler stochastic integrals when the Wiener process is multidimensional

Summary

Introduction

Numerical analysis for stochastic differential equation (SDE) has seen a considerable development in recent years. Davie [4] uses coupling and gives order one for the strong convergence for stochastic differential equations (SDEs). Davie in [17] applied the Vaserstein bound to solutions of vector SDEs and uses the Komlos et al theorem to get order one approximation under a nondegeneracy assumption. The following theorem, which will be stated without proof, gives sufficient conditions for existence and uniqueness of a solution of a stochastic differential equation. Under these conditions ((i)–(iii)) the stochastic differential equation (1) has a unique solution X(t) ∈ [t0, T] with sup E (|X (t)|2) < ∞. Strong Convergence for SDEs. Let (Ω, F, P) be a probability space satisfying the following: Ω is the set of continuous functions with the supremum metric on the interval [0, T], F is the σ-algebra of Borel sets, and P is the Wiener measure.

Numerical Method for Approximating the SDEs

Two-Level Approximation

Numerical Example