Convergence rate of the truncated Milstein method of stochastic differential delay equations

Abstract

This paper is concerned with the strong convergence of highly nonlinear stochastic differential delay equations (SDDEs) without the linear growth condition. On the one hand, these nonlinear SDDEs do not have explicit solutions, therefore implementable numerical methods for such SDDEs are required. On the other hand, the implicit Euler methods are known to converge strongly to the exact solution of such SDDEs. However, they require additional computational efforts. In this article, we propose the truncated Milstein method which is an explicit method under the local Lipschitz condition plus Khasminskii-type condition, study its pth monent boundedness (p is a parameter in Khasminskii-type condition) and show that its rate of strong convergence is close to one.