Some Numerical Methods For Stochastic Differential Equations With Applications

Recently, stochastic differential equations play an important role in science and engineering, including finance, biology, economics and physics. The mathematical modelings of these fields are not fully described by deterministic differential equations, and stochastic differential equations can perfect instead. SDE are based on stochastic calculus, probability theory and stochastic processes. It's very difficult to calculate the analytical solutions of SDE. Therefore, It is very important to construct numerical methods for approximating their solutions.Firstly, the background and the present research of SDE are introduced in the first chapter. The second chapter, some preparatory knowledge of SDE are introduced. Including Brownian motion,stochastic integrals and stochastic Taylor Expansion. Two of the very important forms of SDE, Ito SDE and Stratonovich SDE, deduced by stochastic integrals. In addition, I mention the theorem of necessary and sufficient conditions for existence and uniqueness of a strong solution for SDE and I give the analytical solution of linear SDEs. In order to developing efficient numerical methods, convergence and stability has been considered in the third chapter.The fourth chapter introduced the Euler method and the Milstein method for solving SDE. Including explicit scheme,semi-implicit scheme,implicit scheme and reversed implicit scheme. We analyzed the stability of these scheme, and drawn the corresponding stability regions. In the last of this chapter we have done three numerical experiments. The first numerical experiment confirm the strong convergence step of EM method and Milstein method are 0.5 and 1.0 respectively. The second numerical experiment obtained two results based on one-dimensional linear SDE. The result 1 we get that EM method and Milstein method converge to the solution process of Ito-type SDE under the same Brown way. The result 2 are numerical solutions of the explicit scheme,semi-implicit scheme and reversed implicit scheme of the Euler method and the Milstein method method converge to the true solution. But when diffusion coefficient is big the implicit Euler method does not convergence, however, when diffusion coefficient is small, the solution is convergence. The third numerical based on a two-dimensional stiff stochastic differential equation. Usually even if the definite system, if it has the stiff, solve will also become very difficult. But reversed implicit Euler method and reversed implicit Milstein method can deal with this kind of problem effectively.In the fifth chapter, we apply Predictor-Corrector method to solve implicit scheme for non-linear stochastic differential equations. We also carry out numerical experiments in the last section. We analyzed and compared the average global error of each numerical method.