The spectral collocation method for stochastic differential equations

Can Huang,Zhimin Zhang

doi:10.3934/dcdsb.2013.18.667

Abstract

In this paper, we use the Chebyshev spectral collocation method to solve a certain type ofstochastic differential equations (SDEs). We also use this methodto estimate parameters of stochastic differential equations fromdiscrete observations by maximum likelihood technique and Kesslertechnique. Our numerical tests shows that the spectral method givesbetter results than the Euler's method and the Shoji-Ozaki method.

Highlights

Numerical solutions of stochastic differential equations (SDEs) have caught the attention of researchers for a very long time, a few examples are [9, 10, 15, 18, 19, 20]
Differentiable, Wt is the standard Wiener process. It is well-known that spectral collocation methods gain exponential rate of convergence if the solution of a deterministic equation is analytic and much more accurate than finite difference schemes and Runge-Kutta schemes
Here and hereafter, we only focus on the following stochastic differential equation dXt = b(t, Xt)dt + dWt, X0 = x0

Summary

Introduction

Numerical solutions of stochastic differential equations (SDEs) have caught the attention of researchers for a very long time, a few examples are [9, 10, 15, 18, 19, 20]. We attempt to use the spectral collocation method to solve stochastic differential equations of the form dXt = f (t, Xt)dt + g(Xt)dWt, X(0) = x0. CAN HUANG AND ZHIMIN ZHANG differentiable, Wt is the standard Wiener process It is well-known that spectral collocation methods gain exponential rate of convergence if the solution of a deterministic equation is analytic and much more accurate than finite difference schemes and Runge-Kutta schemes. This is a tractable trick in parameter estimation, the Normal distribution can not be expressed by the original Wiener process, which leads to the failure of strong approximation of the solution process We exclude this method for Example 1.

SPECTRAL METHOD FOR SDE

Conclusion