The Stochastic Differential Equation Truncates The Almost Everywhere Stability Method Of The Euler-Maruyama Method

The convergence and stability of numerical methods for stochastic differential equations with the global Lipschitz continuous coefficients have been well studied.However,most SDE models in real life do not satisfy the global Lipschitz condition.So it is very interesting to study the strong convergence and the stability under the non-global Lipschitz condition.Higham,Mao and Stuart investigated the strong convergence of some numerical schemes under the local Lipschitz and the linear growth condition in 2002,which opened a new chapter in numerical analysis of nonlinear stochastic differential equations.The linear growth condition is still too restrictive,when the drift or the diffusion coefficient is super-linear,we may not deal with the stochastic differential equations by classical numerical methods.And some literatures showed that the explicit Euler-Maruyama may not guarantee the convergence when the coefficients are super-linear.Although the implicit method is feasible to this problem,the computational cost and complexity are relatively high compared with the explicit scheme.That is,the explicit method has less computational cost and simple algebraic structure.Therefore,some modified explicit Euler-Maruyama methods have been developed to solve this problem for improving the convergence and stability,including the tamed Euler-Maruyama method,the stopped Euler-Maruyama method,and the truncated Euler-Maruyama method.Here we mainly consider the truncated Euler-Maruyama method.To make a new numerical method well-defined,we need to consider its convergence and the stability.The strong convergence and its rate have been presented in two of Professor Xuerong Mao's papers for the truncated Euler-Maruyama method.However the stability of the truncated Euler-Maruyama method has not been well studied.Therefore,the main aim of this paper is to study the stability of the scheme under local Lipschitz and Khasminskii type conditions.Although there are two kinds of stochastic stability,named Lpand almost sure stability,the present paper considers the almost sure stability by applying the semi-martingale convergence theorem.A numerical example is given to confirm corresponding theoretical result.