Implicit stochastic Runge-Kutta methods for stochastic differential equations

Kevin Burrage, Tianhai Tian

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28 Citations (Scopus)


In this paper we construct implicit stochastic Rungea??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.
Original languageEnglish
Pages (from-to)21 - 39
Number of pages19
JournalBIT (Copenhagen)
Issue number1
Publication statusPublished - 2004

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