Numerical Solution Of Jump-diffusion Stochastic Differential Equation And Application

In this dissertation we investigate the numerical solution property of the stochastic differential equation(SDE) with jump,numerical simulation and some application on financial calculus.This dissertation consists of three chapters.As follows are main contents.In chapter one,Mainly introduce some basic mathematics knowledge about this area:The Ito formula of SDE with jump,the definition of strong and weak convergence, stochastic Taylor expansions,SDE with jump and some ordinary numerical methods for jump-diffusion SDEs.In chapter two,It is proved that the Euler-Maluyama numerical method has strong convergence rate of at least 1/2,although the proof is given under global Lipschitz condition, one can easily get the same results under a more general non-global Lipschitz condition if we give some restriction on initial data.Further,the split-step backward method is extended to multidimensional case,this method can predigest the numerical computation.In chapter three,Strong Taylor scheme and jump-adapted scheme are introduced .For the need of numerical simulation,some stochastic Taylor expansions are deduced, include strong 1 order Taylor expansion,Milstein-Maghsoodi jump-adapted expansion and so on.Further,this chapter compare several numerical simulation methods though the result of a numerical experiment base on asset pricing which has relation with jumpdiffusion model.Finally,one kind bond pricing is discussed,and it is introduced that how to use the Monte-Carlo method to solve this kind problem.