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SIAM REVIEW 2001 Society for Industrial and Applied Mathematics

Vol. 43 ,No. 3 ,pp. 525–546

An Algorithmic Introduction to

Numerical Simulation of

Stochastic Differential

Equations∗

Desmond J. Higham†

Abstract. A practical and accessible introduction to numerical methods for stochastic differential

equations is given. The reader is assumed to be familiar with Euler’s method for de-

terministic differential equations and to have at least an intuitive feel for the concept of

a random variable; however, no knowledge of advanced probabilit ytheor yor stochastic

processes is assumed. The article is built around 10 MATLAB programs, and the topics

covered include stochastic integration, the Euler–Maruyama method, Milstein’s method,

strong and weak convergence, linear stability, and the stochastic chain rule.

Key words. Euler–Maruyama method, MATLAB, Milstein method, Monte Carlo, stochastic simula-

tion, strong and weak convergence

AMS subject classifications. 65C30, 65C20

PII. S0036144500378302

1. Introduction. Stochastic differential equation (SDE) models play a promi-

nent role in a range of application areas, including biology, chemistry, epidemiology,

mechanics, microelectronics, economics, and finance. A complete understanding of

SDE theory requires familiarity with advanced probability and stochastic processes;

picking up this material is likely to be daunting for a typical applied mathematics

student. However, it is possible to appreciate the basics of ho wto simulate SDEs

numerically with just a background knowledge of Euler’s method for deterministic

ordin