Malliavin and Dirichlet structures for independent random variables

Abstract

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron-Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of U -statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.

Malliavin–Stein method: a survey of some recent developments

“…Further applications of the Malliavin-Stein techniques in the framework of Dirichlet structures are contained in [32,31]. The next section focuses on a discrete Markov structure for which exact fourth moment estimates are available.…”

Continuation value computation using Malliavin calculus under general volatility stochastic process for American option pricing

“…A standard, while restrictive, assumption required to apply the Malliavin calculus is the independence of the increments [9], as the distribution of the increments then depends on the time intervals uniquely. This assumption will be crucial in our developments too, when using stochastic integration by parts.…”

“…Since its invention in 1978, Malliavin calculus [21] have been used in various mathematical fields. In theoretical research, it has been investigated for instance in Lie algebra [19], probability [8,9], stochastic analysis [20]. Recently Laukkarinen [18] combined Malliavin calculus with fractional derivatives.…”

Bounds in $L^1$ Wasserstein distance on the normal approximation of general M-estimators

“…This heuristic has been made rigorous by second-order Poincaré inequalities, which bound distances to the Gaussian when certain functions of the first and second derivatives are small. They have been introduced in the Gaussian setting by Chatterjee [22], extended in [41], and analogues for general independent random variables via discrete secondorder derivatives were studied in [21,26,28]. Second-order Poincaré inequalities for non-Gaussian, non-independent random variables do not seem to have been yet addressed in the literature, and warrant further investigation.…”