Study of Higher Order Split-Step Methods for Stiff Stochastic Differential Equations

Abstract

Stochastic differential equations(SDEs) play an important role in many branches of engineering and science including economics, finance, chemistry, biology, mechanics etc. SDEs (with m-dimensional Wiener process) arising in many applications do not have explicit solutions, which implies the development of effective numerical methods for such systems. For SDEs, one can classify the numerical methods into three classes: fully implicit methods, semi-implicit methods and explicit methods. In order to solve SDEs, the computation of Newton iteration is necessary for the implicit and semi-implicit methods whereas for the explicit methods we do not need such computation. In this thesis the common theme is to construct explicit numerical methods with strong order 1.0 and 1.5 for solving Itˆo SDEs. The five-stage Milstein(FSM)methods, split-step forward Milstein(SSFM)methods and M-stage split-step strong Taylor(M-SSST) methods are constructed for solving SDEs. The FSM, SSFM and M-SSST methods are fully explicit methods. It is proved that the FSM and SSFM methods are convergent with strong order 1.0, and M-SSST methods are convergent with strong order 1.5.Stiffness is a very important issue for the numerical treatment of SDEs, similar to the case of deterministic ordinary differential equations. Stochastic stiffness can lead someone to use smaller step-size for the numerical simulation of the SDEs. However, such issues can be handled using numerical methods with better stability properties. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the FSM and SSFM methods are larger than the Milstein and three-stage Milstein methods. The M-SSST methods possess large mean square stability region as compared to the order 1.5 strong Itˆo-Taylor method. SDE systems simulated with the FSM, SSFM and M-SSST methods show the computational efficiency of the methods. In this work, we also consider the problem of computing numerical solutions for stochastic delay differential equations(SDDEs) of Itˆo form with a constant lag in the argument. The fully explicit methods, the predictor-corrector Euler(PCE)methods, are constructed for solving SDDEs. It is proved that the PCE methods are convergent with strong order γ = ½ in the mean-square sense. The conditions under which the PCE methods are MS-stable and GMS-stable are less restrictive as compared to the conditions for the Euler method.