The Steady-state Distribution Of Stochastic Differential Equations Is Approximated Based On The Non-standard Euler-Maruyama Method

Stochastic differential equations arise in many research fields such as biology and finance.However,since analytical solutions of most stochastic differential equations are difficult to obtain,numerical approximation becomes one of the main research approaches.As with the numerical solution of ordinary differential equations,the stability of the numerical solution of stochastic differential equations has also received more attention.Here we mainly consider that when the time approaches to infinity,the conditions and speed of the numerical solution tends to zero almost surely or in the sense of p-th moment.There is also a special long-time behavior for stochastic differential equations,that is,a steady-state distribution.We need to investigate the steady-state distribution in the sense of convergence in the distribution.The main concern of this paper is how the numerical solution reproduces the steady-state distribution of the analytical solution.At present,the existing research results mainly consider the numerical approximation to the steady-state distribution of stochastic differential equations under the global Lipschitz condition or local Lipschitz condition together with linear growth condition.The numerical methods include the classical Euler-Maruyama method,the ?-method,and so on.On the other hand,the results of the non-standard Euler-Maruyama method show that the numerical method can preserve the domain invariance of the solution in the sense of strong con-vergence.However,the weak convergence result is relatively less,especially its steady-state dis-tribution.Therefore,in this paper,we investigate the steady-state distribution of the non-standard Euler-Maruyama method under local Lipschitz and linear growth condition and its convergence to that of the true solution.We completed the theoretical proof,and then verified the correctness of the theoretical analysis through several numerical examples.