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Anticipating stochastic differential systems with memory

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  • Mohammed, Salah
  • Zhang, Tusheng

Abstract

This article establishes existence and uniqueness of solutions to two classes of stochastic systems with finite memory subject to anticipating initial conditions which are sufficiently smooth in the Malliavin sense. The two classes are semilinear stochastic functional differential equations (sfdes) and fully nonlinear sfdes with a sublinear drift term. For the semilinear case, we use Malliavin calculus techniques, existence of the stochastic semiflow and an infinite-dimensional substitution theorem. For the fully nonlinear case, we employ an anticipating version of the Itô-Ventzell formula due to Ocone and Pardoux [D. Ocone, E. Pardoux, A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations, Annales de l'Institut Henri Poincaré. Probabilité s et Statistiques 25 (1) (1989) 39-71]. In both cases, the use of Malliavin calculus techniques is necessitated by the infinite dimensionality of the initial condition.

Suggested Citation

  • Mohammed, Salah & Zhang, Tusheng, 2009. "Anticipating stochastic differential systems with memory," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2773-2802, September.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:9:p:2773-2802
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    References listed on IDEAS

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    1. Arnold, Ludwig & Imkeller, Peter, 1996. "Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 19-54, March.
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    Cited by:

    1. Ofelia Bonesini & Antoine Jacquier & Chloe Lacombe, 2020. "A theoretical analysis of Guyon's toy volatility model," Papers 2001.05248, arXiv.org, revised Nov 2022.

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