High Strong Order Methods for Non-commutative Stochastic Ordinary Diierential Equation Systems and the Magnus Formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary diierential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the diiculty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage (1996) were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that extant stochastic Runge-Kutta methods suuer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe diiculties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these diiculties by some new techniques. An additional diiculty in solving SODEs arises even in the linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper rst the work of Magnus (1954) (applied to deterministic non-commutative linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed-hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order 1 Runge-Kutta methods with d + 1 stages can be constructed by evaluating a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be ef-ciently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the eecacy of these approaches.