The Random Periodic Solutions And Its EM Numerical Approximation Of A Class Of Stochastic Differential Equations

Random periodic solutions are the behavior of the stochastic process in a longtime or infinite-time domain,which are important issues in the field of stochastic calculus.The concept of random periodic solutions indicates that periodic curves are not pathwise of random dynamical systems,but those systems will move from one periodic curve to another at a particular time.Therefore,the random periodic solutions are composed of infinitely many randomly moving curves in space,which describe a more complex random nonlinear phenomenon than the stationary solutions.Many phenomena can be characterized by dynamic systems,and random periodicity is also common in real phenomena.Studying the random periodic solutions of stochastic differential equations(SDEs)can help us understand the periodic behavior of dynamic systems under internal or external disturbances.The research object of this article is the random periodic solution of a class of semi-linear SDEs.The specific research work is generally as follows:Firstly,the significant assumptions for SDEs are confirmed:the drift coefficient satisfies the one-sided Lipschitz condition,and the diffusion coefficient satisfies the global Lipschitz condition.For the existence and uniqueness of random periodic solutions,we verify the basic properties of the solutions of SDEs with common tools in stochastic calculus such as the It? formula,Gronwall inequality and find the relationship between the solution and the initial value.Then using these conclusions and the properties of the semi-flow,we verify the existence and uniqueness of the random periodic solutions.Based on the proof of the existence and uniqueness,a numerical approximation of the random periodic solutions are subsequently considered.To solve this problem,this paper introduces the tamed Euler Maruyama(EM)scheme to ensure that the constructed drift coefficients satisfy moment boundness under system assumptions.The properties of numerical solutions are discussed in the infinite time domain[-kτ,0)and finite time domain[0,t],respectively.It is further proved that the tamed EM numerical solutions of the random periodic solutions converge to the exact solutions,revealing the convergence rate between the exact random periodic solutions and the approximated one is α∈(0,1/2)in the L2.Finally,we give an example to simulate the approximated random periodic solutions.