Standardizing densities on Gaussian spaces

Lanconelli, Alberto

doi:10.1016/j.spl.2018.01.033

Cited by 1 publication

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References 18 publications

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“…We also remark that, in addition to its key role in the theory of Itô-Skorohod integration (1.7), the Wick product has important probabilistic interpretations in the Gaussian and Poissonian analysis. See [17], [18], [16] and the references quoted there.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

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

Lanconelli

2020

Journal of Mathematical Analysis and Applications

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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 .

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Section: Proof Of Theorem 14mentioning

confidence: 99%