The Truncated EM Method For Stochastic Differential Equations Driven By Continuous Semimartingales

As a new mathematics subject,stochastic differential equation has been widely applied in many fields such as biology,economy and engineering because of its wide application prospect.More and more scholars have devoted themselves to study the stochastic differential equation,thus the theory of SDE were developed.The existence and uniqueness of solutions of stochastic differential equations,the numerical approximation method and the properties of the solution(convergence,stability,etc.)are still hot topics studied by scholars,and its research has important theoretical significance and practical value.In 2015,Professor Mao proposed the truncated EM method for non-linear stochastic differential equations,and hence established the strong convergence theory in finite time.In this dissertation,the stochastic differential equation driven by the continuous semimartingales is investigated by the truncated EM method.The stochastic differential equation driven by Brownian motion is an ideal state,which has certain limitation in practical application.Therefore this paper extends it to the stochastic differential equation driven by continuous semimartingales.In general,global Lipschitz conditions guarantee the existence of global solutions,and linear growth conditions guarantee uniqueness of solution.However,many equations only satisfy the local Lipschitz condition which cannot guarantee the existence of global solution.In this paper,we mainly study the stochastic differential equation driven by continuous semimartingales under the local Lipschitz condition plus the Khasminskii-type condition.Not only proves the existence and uniqueness of the solution,but also gives the convergence of the truncated EM method.By the knowledge of random integration,we first prove the pathwise uniqueness,then construct a solution of the equation,and finally obtain the existence and uniqueness of the solution.By the stopping time technology,Gronwall inequality and so on,we proved the strong convergence of the truncated EM numerical solution.