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

• Yaozhong Hu ^[1] ; Salah-Eldin A. Mohammed ^[2] ; Feng Yan ^[3]
  1. [1] University of Kansas

     University of Kansas

     City of Lawrence, Estados Unidos

     
  2. [2] Southern Illinois University
  3. [3] Williams Energy

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 1, 1, 2004, págs. 265-314
• Idioma: inglés
• DOI: 10.1214/aop/1078415836
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.