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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:482

Article

Lanconelli, Alberto¹ 

[1] Alma Mater Studiorum Univ Bologna, Dipartimento Sci Stat Paolo Fortunati, Via Belle Arti 41, I-40126 Bologna, Italy

关键词: Stochastic differential equations;    Wong-Zakai approximation;    Malliavin calculus;   

DOI  :  10.1016/j.jmaa.2019.123557

来源: Elsevier

【 摘 要 】

We investigate the regularity of the law of Wong-Zakai-type approximations for Ito stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using 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 Ito 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free