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Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations

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  • Lanconelli, Alberto
  • Scorolli, Ramiro

Abstract

We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô’s SDEs, we consider for any finite partition of the time interval [0,T] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero.

Suggested Citation

  • Lanconelli, Alberto & Scorolli, Ramiro, 2021. "Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 57-78.
    • Handle: RePEc:eee:spapps:v:141:y:2021:i:c:p:57-78
    DOI: 10.1016/j.spa.2021.07.007
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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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