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January 2004 Discrete-time approximations of stochastic delay equations: The Milstein scheme
Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan
Ann. Probab. 32(1A): 265-314 (January 2004). DOI: 10.1214/aop/1078415836


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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Yaozhong Hu. Salah-Eldin A. Mohammed. Feng Yan. "Discrete-time approximations of stochastic delay equations: The Milstein scheme." Ann. Probab. 32 (1A) 265 - 314, January 2004.


Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1062.60065
MathSciNet: MR2040783
Digital Object Identifier: 10.1214/aop/1078415836

Primary: 34K50 , 60H07 , 60H35
Secondary: 34K28 , 37H10 , 60C30 , 60H10

Keywords: anticipating calculus , Itô's formula , Malliavin calculus , Milstein scheme , tame functions , weak derivatives

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 1A • January 2004
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