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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.

Keywords

Stochastic differential equations Wong–Zakai theorem Wick product 

Mathematics Subject Classification (2010)

60H10 60H30 60H05 

Notes

Acknowledgements

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

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Tunis - El ManarNabeulTunisia
  2. 2.Dipartimento di MatematicaUniversitá degli Studi di Bari Aldo MoroBariItaly

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