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Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations

Author

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  • Bilel Kacem Ben Ammou

    (University of Tunis - El Manar)

  • Alberto Lanconelli

    (Universitá degli Studi di Bari Aldo Moro)

Abstract

We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Itô equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Itô equations solve approximated Stratonovich equations with a certain correction term in the drift.

Suggested Citation

  • Bilel Kacem Ben Ammou & Alberto Lanconelli, 2019. "Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1780-1803, December.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:4:d:10.1007_s10959-018-0837-x
    DOI: 10.1007/s10959-018-0837-x
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    References listed on IDEAS

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    1. Da Pelo, Paolo & Lanconelli, Alberto & Stan, Aurel I., 2013. "An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3183-3200.
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