Two Classes Of Numerical Methods For Solving Stochastic Differential Equations

With the spreed of technology and science, both in the nature and the real life, theeffect of uncertain factors is nonnegligible to the development state of things. Hence itis more precised to portray the motion law of the nature by using stochastic differentialequations. But stochastic differential equations usually have strong the coupling andnonlinear property, it is difficult to calculate the expression of the solution, so we needto construct numerical methods to approximate the solutions of stochastic differentialequations, and further simulate the behavior of solutions by using the mathematical tooland computer. In order to simulate the quantitative behavior of solutions moreaccurately, we require the constructed numerical methods to maintain the structure ofthe original stochastic system, such as symplectic structure, phase volumes and firstintegrals, etc.Firstly, our work introduces the research background of stochastic differentialequations with its established purpose and significance. Then it summarizes thedevelopment status for numerical method of stochastic differential equations, bothdomestic and overseas.Secondly, we briefly introduces the required preliminary knowledge.Moreover, we construct two types of non-standard finite difference method whichbased on Euler-Maruyama method and Milstein method. By comparing the constructednon-standard finite difference method of one-step approximation and the deviation ofthe exact solution of the original equation, we obtain two types of non-standard finitedifference method with the mean-square convergence order, and apply numericalexamples to verify the theoretical results.Finally, for a special class of stochastic differential equations, this dissertationutilizes the stochastic Runge-Kutta methods to calculate the equations and deducesstochastic Runge-Kutta-Nystr(?)m methods. The numerical solution and the exactsolution will be expanded as the stochastic Taylor expansion. By comparing the errororder of the expansions, we get order conditions of1.0order strong convergence ofstochastic Runge-Kutta-Nystr(?)m methods. In addition, this paper uses stochasticRunge-Kutta-Nystr(?)m method to solve stochastic Hamilton systems, analyzessymplectic conditions of stochastic Runge-Kutta-Nystr(?)m methods, and proves that thestochastic symplectic Runge-Kutta-Nystr(?)m method maintain the original system ofquadratic conservation quantity accurately. At last, this paper solves several practicalproblems, and compares the stochastic symplectic Runge-Kutta-Nystr(?)m method andnonsymplectic numerical method by simulating in a long time. Then this paperconcludes the advanced property of stochastic symplectic Runge-Kutta-Nystr(?)mmethod that has been constructed.