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

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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.

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Acknowledgements

The author acknowledges the support of the Italian INDAM-GNAMPA.

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Authors and Affiliations

  1. Department of Mathematics, University of Tunis - El Manar, Street Mohamed Alaya Kacem, Nabeul, Tunisia

    Bilel Kacem Ben Ammou

  2. Dipartimento di Matematica, Universitá degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125, Bari, Italy

    Alberto Lanconelli

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  1. Bilel Kacem Ben Ammou
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  2. Alberto Lanconelli
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Correspondence to Alberto Lanconelli.

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Ben Ammou, B.K., Lanconelli, A. Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations. J Theor Probab 32, 1780–1803 (2019). https://doi.org/10.1007/s10959-018-0837-x

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  • DOI: https://doi.org/10.1007/s10959-018-0837-x

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