Asymptotic moment boundedness of the numerical solutions of stochastic differential equations

Few papers look at the asymptotic boundedness of numerical solutions of stochastic differential equations (SDEs). One of the open questions is whether numerical approximations can reproduce the boundedness property of the underlying SDEs. In this paper, we give positive answer to this question. Firstly we discuss the asymptotic moment upper bound of the Itô type SDEs and show that the Euler–Maruyama (EM) method is capable to preserve the boundedness property for SDEs with the linear growth condition on both drift and diffusion coefficients. But under the weaker assumption, the one-sided Lipschitz, on the drift coefficient, the EM method fails to work. We then show that the backward EM method can work in this situation.