Implicit stochastic Runge-Kutta methods for stochastic differential equations

Burrage, K. and Tian, T. (2004) Implicit stochastic Runge-Kutta methods for stochastic differential equations. BIT Numerical Mathematics, 44(1), pp. 21-39. (doi: 10.1023/B:BITN.0000025089.50729.0f)

Abstract

In this paper we construct implicit stochastic Runge–Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Tian, Dr Tianhai
• Authors: Burrage, K., and Tian, T.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: BIT Numerical Mathematics
• ISSN: 0006-3835
• ISSN (Online): 1572-9125
• Published Online: 28 October 2004