Convergence And Stability Of Numerical Solutions For Stochastic Differential Equations With Poisson Measure

As mathematical models, stochastic differential equations with Poisson measurehave important theoretical value and practical application significance in physics, chem-istry, biology, medicine, geology, astronomy, finance and other fields. This kind of e-quations are often used to establish the models for the discontinuous tracks of Markovprocesses, especially to describe the operation of financial markets in the financial cri-sis. Since it’s difficult to obtain the exact solutions of these equations, the study of thenumerical methods and properties is of great importance in theory and application. Thispaper mainly deals with the convergence and stability of numerical solutions for stochas-tic differential equations with Poisson measure.This paper first presents the application background and the research history of s-tochastic differential equations with Poisson measure, and then reviews the basic proper-ties of the exact solutions and the development of almost sure stability and p th (p>2)moment stability. The current situation of numerical solutions to stochastic differential e-quations with Poisson measure is also presented, together with commonly used notationsand basic knowledge.Implicit compensated Euler method is constructed by using the compensated mea-sure of Poisson measure for the d dimensional stochastic differential equations with Pois-son measure. Mean-square convergence of the implicit compensated Euler method isproved under the global Lipschitz conditions and the linear growth conditions. Mean-square exponential stability of the equations is derived under the one-sided Lipschitz con-ditions. On the premise of ensuring the equations to be exponentially stable in meansquare, for arbitrary selected step size, the implicit compensated Euler method is provedto be exponentially stable in mean square. With the numerical examples, the conclusionsof the convergence and stability are verified by Matlab drawing.The Euler method is given for the d dimensional stochastic differential equationswith Poisson measure. It is proved that the exact solutions and numerical solutions of theequations are in a compact set with a large probability under the non-Lipschitz conditions.The convergence in probability of the Euler method is given under the non-Lipschitzconditions and then is verified by a numerical example. The Euler method is constructed for the d dimensional stochastic delay differentialequations with Poisson measure. The global solutions for the equations are given andproved to be unique under the generalized Khasminskii-type conditions. It is proved thatthe exact solutions and numerical solutions of the equations are in a compact set with alarge probability under the generalized Khasminskii-type conditions. The convergencein probability of the Euler method is verified. The corresponding numerical example isgiven for the convergence.Finally, by using the compensated measure of Poisson measure, implicit compensat-ed Euler method is constructed for the d dimensional equations with piecewise continuousarguments driven by Wiener process and Poisson measure. The concept of the exact solu-tions of the equations is defined. The existence and uniqueness of the exact solutions areproved under the global Lipschitz conditions and the linear growth conditions. The con-vergence of the implicit compensated Euler method is verified under the global Lipschitzconditions and the linear growth conditions. Under the one-sided Lipschitz conditons,the mean-square asymptotical stability is given for the equations and it is proved that theimplicit compensaed Euler method is mean-square asymptotically stable.