We present a review of the most important historical as well as recent results of Malliavin calculus in the framework of the Wiener-Itô chaos expansion. AMS Mathematics Subject Classification (2010): 60H40, 60H07, 60H10, 60G20

We translate in semi-group theory our proof of Varadhan estimates for subelliptic Laplacians which was using the theory of large deviations of Wentzel-Freidlin and the Malliavin Calculus of Bismut type. Key–Words: Large deviations. Subelliptic estimates.

We establish the hypoellipticity of a large class of highly degenerate second order di erential operators of H ormander type. The hypotheses of our theorem allow Hormander's general Lie algebra condition to fail on a collection of hypersurfaces. The proof of the theorem is probabilistic in nature. It is based on the Malliavin calculus and requires new sharp estimates for di usion processes in...

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic setting some recent results by Nualart and Peccati on characterizations of ...

The aim of this paper is extensively investigate the performance of the estimators for the Greeks of multidimensional complex path-dependent options obtained by the aid of Malliavin Calculus. The study analyses both the computation effort and the variance reduction in the Quasi-Monte Carlo simulation framework. For this purpose, we adopt the approach employed by Montero and Kohatsu-Higa to the ...

We study in this paper weak approximations Wasserstein-1 distance to stochastic variance reduced gradient Langevin dynamics by delay differential equations, and obtain uniform error bounds. Our approach is via Malliavin calculus a refined Lindeberg principle.

<p style='text-indent:20px;'>By using distribution dependent Zvonkin's transforms and Malliavin calculus, the Bismut type formula is derived for intrinisc/Lions derivatives of SDEs with singular drifts, which generalizes corresponding results classical regular SDEs.</p>

For general time-dependent local volatility models, we propose new approximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. ...

By using Bismut’s approach to the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Lévy noises. Under full Hörmander’s conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density. Moreover, under a uniform first order Lie’s bracket condition, we...