Numerical Study Of Stochastic Differential Equations

This thesis consists of two parts. The first one is devoted to strong convergence and stability of numerical methods for finite dimensional stochastic ordinary differential equations (SODEs). More precisely, for stiff SODEs and SODEs with non-globally Lipschitz continuous coef-ficients, we respectively design different numerical schemes and inves-tigate their convergence and stability properties. The second part is concerned with strong approximations to stochastic partial differential equations (SPDEs). In particular, a new Runge-Kutta type method is constructed for semi-linear SPDEs with multiplicative trace class noise. Compared with existing Euler-type and Milstein-type schemes, the new scheme needs less computational efforts.In the first chapter, the research background is introduced on numerics of SODEs and SPDEs. Moreover, the history and current development of the research topic are briefly reviewed. Finally, we make an outline of the main research work of this thesis.In Chapter2, elementary basis of stochastic analysis in Hilbert space is presented briefly. Besides, several martingale inequalities are given, which are used in later development.Chapter3is concerned with fully implicit numerical methods for stiff SODEs and a new family of fully implicit Milstein methods are here constructed. Further, the new methods are shown to be strongly convergent to a class of SODEs, with strong convergence order one. Also, both linear mean-square and almost sure asymptotic stability of such methods are investigated.In Chapter4, a new class of compensated stochastic θ methods are proposed for SODEs with Poisson jumps and their strong convergence and mean-square stability are examined. Compensated stochastic θ methods applied to mean-square stable linear scalar test equation are mean-square stable for any stepsize h>0as1/2≤θ≤1. This property can be regarded as an extension of A-stability for deterministic θ-methods.Chapter5focuses on explicit numerical methods for SODEs with non-globally Lipschitz continuous coefficients. Owing to strong and weak divergence of explicit Euler-Maruyama method and explicit Mil-stein method in non-globally Lipschitz setting, a tamed Euler method has been proposed recently. Here the tamed Milstein method is in-troduced, which has higher strong convergence order than the tamed Euler method does. To be precise, the strong convergence order of one is obtained for the tamed Milstein method in non-globally Lipschitz setting.In the sixth chapter, numerical methods for strong approxima-tions to SPDEs are considered and a new Runge-Kutta type method is constructed for semilinear parabolic SPDEs with multiplicative trace class noise. The new scheme can be regarded as an infinite dimen-sional analog of Runge-Kutta method for finite dimensional SODEs. What is more, its strong convergence is established and a compara-tive study of the new method and existing methods shows the new method's advantage in computational effectiveness.