Stochastic Differential Equations With Variable Step Length Numerical Algorithm Research

Stochastic differential equations(SDEs) arise widely in physics, engineering, biology, medical science, economics and so on. Yet, the number of instances where the analytical solution can be found, is very limited. Hence, investigating more efficient and stable numer-ical methods for SDEs is of great importance to numerous appli-cations, as well as theoretical interests. Four families of stochastic numerical methods are developed in this thesis:adaptive stochastic Runge-Kutta methods(RKMs), Runge-Kutta-type predictor-corrector methods(RK-type PC), adaptive predictor-corrector methods and adap-tive balanced methods. Special attention is paid to the applications of the methods to stiff SDEs. The thesis is organized as follows:In the first Chapter, many applications of stochastic differential equations in different fields are presented, and the development of the numerical methods for SDEs is reviewed, with special attention paid to stiff SDEs.Secondly, background materials for this paper are outlined, main-ly involving probability theory, stochastic process, stochastic differen-tial equations and the core concepts in the study of numerical solutions of SDEs.In Chapter3, a new adaptive explicit RKMs is developed and the corresponding mean-square stability property is investigated. The numerical experiments bright its efficiency in the context of non-stiff SDEs, yet its power seems to be limited with respect to the stiff SDEs.Based on the order conditions for RK-type PC methods, a new RK-type PC method, with better results than those of the method proposed by Burrage and Tian[l] in terms of principal error coeffi-cients, is presented in the fourth chapter. Chapter5consists of two families of adaptive RK-type PC meth-ods:the first one with strong order1.0, based on the methods de-veloped in chapter4; the second one with strong order1.5, based on the adaptive methods introduced in chapter3and an implicit Runge-Kutta methods with strong order1.5. They fit better to stiff SDEs than the underlying methods.Finally, two adaptive balanced methods, balanced Euler and bal-anced Milstein methods, are proposed. The numerical experiments show they are very promising in solving stiff SDEs numerically with better efficiency and accuracy.Adaptive method is one of the strategies to solve SDEs numer-ically with high efficiency. Variable stepsize is a popular technology to develop adaptive methods. It is predictable that the methodology in the thesis will play a role in the research of numerical solutions of SDEs.