Absolute continuity and Fokker-Planck equation for the law of Wong-Zakai approximations of Itô's stochastic differential equations

Abstract

We investigate the regularity of the law of Wong-Zakai-type approximations for Itô stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using a criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density. The parabolic smoothing effect typical of the solutions of Itô equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial condition of the random differential equation needs to possess a density itself. AMS 2000 classification: 60H10; 60H07; 60H30converges, as ε goes to zero, to the solution of the Stratonovich stochastic differential equation (SDE, for short)instead to the more popular Itô's interpretation of the corresponding stochastic equation, dX t = b(t, X t )dt + σ(t, X t )dB t .

Wong-Zakai approximations for quasilinear systems of Itô's type stochastic differential equations

“…This establishes a further similarity between the Wong-Zakai approximating equation (1.11) and its exact counterpart (1.12). This theorem generalizes the one obtained in [10] for the scalar problem (1.3). The proof is postponed to Section 4.…”