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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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.

Article information

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

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1078415836

Digital Object Identifier
doi:10.1214/aop/1078415836

Mathematical Reviews number (MathSciNet)
MR2040783

Zentralblatt MATH identifier
1062.60065

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aop/1078415836


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