Stochastic Differential Equations And Its Numerical Methods

In many fields, stochastic differential equations (SDEs) are more and more im-portant in mathematical models, such as, in financial systems, quantitative, control systems, statistical physics, biology systems and so on. However, due to the lack of a effective method to solve SDEs, the stochastic factors contained in these fields were often been ignored, which brings some essential restriction in practical applications. Recently, with the development of modern computer, some stochastic models can be studied via the simulation of high speed computer. But, because of the complex-ity and diversity of stochastic systems, except some special SDEs, in general it is hard to obtain the explicit solution for a general given SDEs ([11]). Consequently, characterizing the property of the moments is a proper and helpful method for under-standing SDEs, and then constructing stochastic numerical methods is particularly important.In the first two chapters, we present some basic theories and background about SDEs, including Brown motion, a glimpse of stochastic analysis, the existence-uniqueness theorem, linear SDEs and its solutions'explicit expression.In Chapter 3, we first present some definitions and basic results about the convergence and stability of numerical methods, then we prove that the 2-stage R-K method is strong 1.0 convergent by selecting the appropriate matrix A, B and vectorα,β. The main purpose of this chapter is to discuss the stability of the three forms (explicit, semi-implicit and implicit) for R-K methods. We first deduce their MS stable functions, then we obtain the stable domain for both R-K explicit method and R-K semi-implicit method. Based on these results, we show that the stability of R-K explicit method and R-K semi-implicit method are un-comparable, which is different from the corresponding property of Euler method and Milstein method.In Chapter 4, using linear differential equation as the test equation, we compare the global error and convergent rate of the three forms of R-K method, Euler method and Milstein method; and we present our compared results via some visual graphs and tables. These results show explicitly that R-K methods is the best one if we only consider the MS stability. Finally, we present the approximation of exactly solution and the three forms of the three methods through 9 graphs. Through these visual graphs we can see explicitly that the three forms of R-K method are more approach to the exactly solution than the corresponding forms of Euler and Milstein methods, as well as accuracy and convergence.