Discrete-Time Approximations of Stochastic Delay Equations: The Milstein Scheme, The

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinitedimensional Itô formula for “tame ” functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the nonanticipating nature of the SDDE, the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel. 1. Introduction. Discrete-time strong approximation schemes for stochastic ordinary differential equations (SODEs) are well developed. For an extensive study of these numerical schemes, one may refer to [17], [18] and [19], Chapters 5 and 6. Some basic ideas of strong and weak orders of convergence are illustrated in [13]