Numerical Analysis Of Stochastic Differential Equation Driven By Small Noise Under Non-global Lipschitz Condition

With the development of economic society and great progress made in science and technology, stochastic differential equations (SDEs) models play a more and more important role in a range of application fields, including biology, chemistry, physics, medical, engineering, economic, mathematical finance and so on. However, like ordinary differential equations (ODEs), the analytical solutions of stochastic differential equations are also usually hardly available. So investigating efficient numerical methods for stochastic differential equations is of great importance to numerous applications and theoretical interests. And the development of computer technology also makes it possible to simulate the numerical methods through computer. In this thesis, we study a family of stochastic differential equations (SDEs), i.e., stochastic differential driven by small noise and investigate numerical solvers for such SDEs under non-global Lipschitz condition. The thesis is organized as follows:Firstly, many applicable backgrounds of SDEs with small noise are presented. Numerical methods under non-global Lipschitz condition and numerical methods for small noise problem, which develop in recent years, are also reviewed in this chapter. Our works of the thesis are outlined in the end.In the following chapter, some preliminary materials for the thesis are given, mainly involving probability theory, stochastic analysis and the key concepts in the study of numerical solution of SDEs.Chapter3investigates convergence of stochastic split-step one-leg theta methods when applied to SDEs with small noise under non-global Lipschitz condition. The strong convergence analysis and global error estimates are given. Numerical experiments are carried out to confirm the theoretical findings.In Chapter4, we have analyzed the strong convergence of a family split linear theta methods for SDEs under non-global Lipschitz condition and the global error estimate for such methods applied to SDEs with small noise is also presented. Simulation results illustrate the conclusions. Particularly when the methods are applied to SDEs with small noise, the accuracies are greatly improved.In the last chapter, we give a summation for this thesis. We also outline some areas we want to explore in the future at last.