The Annals of Probability

Discrete-time approximations of stochastic delay equations: The Milstein scheme

Yaozhong Hu, Salah-Eldin A. Mohammed, and Feng Yan

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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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Ann. Probab., Volume 32, Number 1A (2004), 265-314.

First available in Project Euclid: 4 March 2004

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Zentralblatt MATH identifier

Primary: 34K50: Stochastic functional-differential equations [See also , 60Hxx] 60H07: Stochastic calculus of variations and the Malliavin calculus 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 60C30 60H10: Stochastic ordinary differential equations [See also 34F05] 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 34K28: Numerical approximation of solutions

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


Hu, Yaozhong; Mohammed, Salah-Eldin A.; Yan, Feng. Discrete-time approximations of stochastic delay equations: The Milstein scheme. Ann. Probab. 32 (2004), no. 1A, 265--314. doi:10.1214/aop/1078415836.

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