General Order Conditions for Stochastic Runge-kutta Methods for Both Commuting and Non-commuting Stochastic Ordinary Diierential Equation Systems

In many modelling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary diierential equation (SODE). However, unlike the de-terministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by Burrage and Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But Burrage and Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or It^ o) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known Stochastic Runge-Kutta methods can suuer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation.