Strong Convergence Of The Tamed Euler Method Of Stochastic Differential Equations

This paper mainly studies the strong convergence and the convergence rate of the Tamed Euler method of stochastic differential equations(SDEs).Stochastic differential equations have widely applications in many fields such as physics,biology,control science and industry.It is difficult to obtain true solutions of these equations,so the study of numerical solutions is of important theoretical consequences and applicable value.Several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations under the local Lipschitz condition.These numerical methods include the Tamed Euler-Maruyama(EM)method,the Tamed Milstein method,th e stopped EM,etc.The paper mainly studies the Tamed Euler method under the more general conditions.The paper first presents the background of appli cation and research history of stochastic differential equations and their numerical solutions.Secondly,the paper studies the strong convergence of the Tamed Euler method of stochastic differential equations.By introducing the discrete format of the Tamed Euler method,the Tamed Euler method of stochastic differential equations is strongly convergent under the local Lipschitz condition and Khasminskii-type condition.Finally,the paper investigates the convergence rate of the Tamed Euler method of stochastic differential equations.On the one hand,we study the convergence rate of the numerical method at time T in the same condition,and prove that convergence rate is 1/2.On the other hand,we illustrate that Tamed Euler converges at a rate of 1/2 in a finite time interval.The numerical results illustrate that the numerical solution of the Tamed Euler method which can be sufficiently small,approximated by the true solution.In addition Tamed Euler method converges and converges to 1/2.