Numerical methods for strong solutions of stochastic differential equations: An overview. (English) Zbl 1048.65004

Burrage, K.; Burrage, P. M.; Tian, T.

Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460, No. 2041, 373-402 (2004).

Summary: 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. 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, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes.
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 that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)
34F05	Ordinary differential equations and systems with randomness
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

stochastic differential equations; strong solutions; numerical methods; survey paper; trajectories; sample paths; stochastic Taylor series expansion; convergence; Magnus expansion; explicit and implicit methods; Brownian path; stochastic integrals; variable-step-size implementations