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. 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].If the rate of change of a physical system depends only on its present state and some noisy input, then the system can often be described by a stochastic ordinary differential equation (SODE). However, in many physical situations the rate of change of the state depends not only on the present but also on the past states of the system. In such cases, stochastic delay differential equations (SDDEs) or stochastic functional differential equations (SFDEs) provide important tools to describe and analyze these systems. For various aspects of the qualitative theory of SFDEs the reader may refer to [20,21] and the references therein.SDDEs and SFDEs arising in many applications cannot be solved explicitly. Hence, one needs to develop effective numerical techniques for such systems. Depending on the particular physical model, it may be necessary to design strong L p (or almost sure) numerical schemes for pathwise solutions of the underlying SFDE. Strong approximation schemes for SFDEs may be used to

For a given stochastic process X, its segment X t at time t, represents the "slice" of each path of X over a fixed time-interval t − r t , where r is the length of the "memory" of the process. Segment processes are important in the study of stochastic systems with memory (stochastic functional differential equations or SFDEs). The main objective of this paper is to study nonlinear transforms of segment processes. Toward this end, we construct a stochastic integral with respect to the Brownian segment process. The difficulty in this construction is the fact that the stochastic integrator is infinite dimensional and is not a (semi)martingale. We overcome this difficulty by employing Malliavin (anticipating) calculus techniques. The segment integral is interpreted as a Skorohod integral via a stochastic Fubini theorem. We then prove Itô's formula for the segment of a continuous Skorohod-type process and embed the segment calculus in the theory of anticipating calculus. Applications of the Itô formula include the weak infinitesimal generator for the solution segment of a stochastic system with memory, the associated Feynman-Kac formula, and the Black-Scholes PDE for stock dynamics with memory.

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