A Survey Of Weak Approximations Of Stochastic Differential Equations With Jumps

The development of numerical methods for stochastic differential equations has intensified over the past decade.Nevertheless there exists a wide gap between the well developed theory of stochastic differential equations and its application.The crucial task in bridging this gap is the development of efficient numerical methods,which obviously should be implementable on modem digital computers.In view of the need to simulate a large number of different sample paths in order to estimate various statistical features of the solutions,vector or super computers will play an rapidly increasing role here.Researchers have increasingly been studying models from economics and the natural sciences where the underlying randomness contains jumps.To give an example like them,one would like to represent them by a process that has jumps.This paper is a survey of weak approximation of stochastic differential equations with jumps.In this paper,we firstly review the definition and classification of stochastic process.Then,we introduce the second-order moment stochastic variables.Based on this,we present the convergence and integration in the mean. After we introduce the Ito formular and stochastic integration when there are jumps,we describe a number of types stochastic differential equations that involve jumps.Chapter 4 is a description of weak approximations of stochastic differential equations with jumps.As the usual SDEs, the first section presents the convergence law,i.e.strong convergence and weak convergence.In section 2,we introduced the Ito-Taylor expansion.Then we get some weak approximations by appropriate truncation.In the last section in chapter 4,based on the error expansion,we get the extrapolation of the Euler scheme.In the remainder of this paper,we compare the weak approximations that appear before and conclude some results.Although the Euler scheme is very efficient to implement on the computer,the accuracy is not always good after a long time. The EE2 scheme is also very efficient and usually has a better accuracy.In theory,both the EE2 scheme and the WST2 scheme are of second order.Sometimes the EE2 scheme will achieve a better precision than the WST2 scheme,while on the other hand,the WST2 scheme will achieve a better precision than the EE2 scheme,though the WST2 scheme is not very efficient by means of the implementation and the CPU time.It is very interesting to note that the weak Milstein scheme,which has more contributed terms than the Euler scheme,has a lower accuracy than the Euler scheme.This phenomenon also happens in the diffusion case.