Discrete-time approximations of stochastic delay equations: The Milstein scheme
Yaozhong Hu, Salah-Eldin A. Mohammed, and Feng Yan
Source: Ann. Probab. Volume 32, Number 1A
(2004), 265-314.
Abstract
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 infinite-dimensional 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.
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Keywords: Milstein scheme; Itô's formula; tame functions; anticipating calculus; Malliavin calculus; weak derivatives
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1078415836
Digital Object Identifier: doi:10.1214/aop/1078415836
Mathematical Reviews number (MathSciNet): MR2040783
Zentralblatt MATH identifier: 02100719
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