Two Types Of Numerical Methods For The Steady-state Distribution Of Stochastic Differential Equations

In the paper,the stationary distribution of the numerical solution generated by two methods is studied.The first method is the stochastic?method,and the second one is the stochastic Milstein method.The existence and uniqueness of the stationary distribution of the numerical solution generated by the stochastic?method is studied.When the parameter theta takes different values,the requirements on the drift and diffusion coefficients are different.If??[1/2,1],the diffusion coefficient of stochastic differential equations needs to satisfy the global Lipschitz condition,while the drift coefficients need to satisfy the one-sided Lipschitz condition.If??[0,1/2),the drift coefficients and diffusion coefficient of stochastic differential equations needs to satisfy the global Lipschitz condition.Then the convergence of the numerical stationary distribution to the true counterpart is investigated.Several numerical experiments are presented to demonstrate the theoretical results.The Milstein method is used to approximate stationary distribution of stochastic differential equations with commutative noise.When the drift and diffusion coefficients of stochastic differential equations with commutative noise satisfy the global Lipschitz condition,it can be proved that there exists a unique stationary distribution for stochastic differential equation with commutative noise.The decay rate of the transition probability kernel generated by the Milstein method to the unique invariant measure of the method is observed to be exponential with respect to the time variable.The convergence rate of the numerical invariant measure to the underlying counterpart is shown to be one.Numerical simulations are presented to demonstrate the theoretical results.