Study On Numerical Methods For Solving Stochastic Differential Equations Based On Stochastic Taylor Expansion

Stochastic differential equations relate to ordinary differential equations, probability theory, stochastic processes and stochastic analysis etc., and have widely applications in many fields to more and more important in representing the phenomena in real life. However it is very difficult to find the analytical solution of stochastic differential equations, so we need to give effective numerical methods to for solving stochastic differential equations.Chapter1and Chapter2in this thesis briefly introduce the background and research status of stochastic differential equations, the basic concepts and knowledge about stochastic process and stochastic differential equations.Chapter3gives an introduction about the stochastic Taylor expansion, makes a summary of some common lower order numerical methods for solving stochastic differential equations. Also the stability and convergence of numerical solution to stochastic differential equations and some common used inequalities are presented in this chapter.Chapter4gives a numerical method for slove Ito type stochastic differential equations, LZ method, based on the stochastic Taylor expansion, prove the mean square stability of this method, and obtains that the order of its local convergence in mean, the order of its local convergence in mean square and the order of its strong convergence in mean square are all1. The numerical experiment shows that the method introduced in this thesis is more accurate than Euler method and Milstein method.Chapter5presents a modification of LZ method given in Chapter4, LZW method, proves the mean square stability and T-stability of this method and the equivalence between the mean square stability and the exponential stability for this method, and obtains the order of its local convergence in mean is2, the order of its local convergence in mean square is1, and the order of its strong convergence in mean square is1. The numerical experiment intuitive shows that the method has the same accuracy as LZ method.