Numerical Methods for Strong Solutions of Stochastic Differential Equations: an Overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations (SDEs). We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications (section 1), and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes (section 2). In section 3 we present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods which preserve the underlying structure of the problem. In sections 4 and 5 we present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, in section 6 we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable stepsize implementations based on various types of control.