An invariance principle for stochastic series I. Gaussian limits

We study invariance principles and convergence to a Gaussian limit for stochastic series of the form $S(c,Z)=\sum_{m=1}^{\infty }\sum_{\alpha _{1}<...<\alpha _{m}}c(\alpha _{1},...,\alpha _{m})\prod_{i=1}^{m}Z_{\alpha _{i}}$ where $Z_{k}$, $k\in \mathbb{N}$, is a sequence of centred independent random variables of unit variance. In the case when the $Z_{k}$'s are Gaussian, $S(c,Z)$ is an element of the Wiener chaos and convergence to a Gaussian limit (so the corresponding nonlinear CLT) has been intensively studied by Nualart, Peccati, Nourdin and several other authors. The invariance principle consists in taking $Z_{k}$ with a general law. It has also been considered in the literature, starting from the seminal papers of Jong, and a variety of applications including $U$-statistics are of interest. Our main contribution is to study the convergence in total variation distance and to give estimates of the error

Regularity of probability laws by using an interpolation method

We study the problem of the existence and regularity of a probability density in an abstract framework based on a "balancing" with approximating absolutely continuous laws. Typically, the absolutely continuous property for the approximating laws can be proved by standard techniques from Malliavin calculus whereas for the law of interest no Malliavin integration by parts formulas are available. Our results are strongly based on the use of suitable Hermite polynomial series expansions and can be merged into the theory of interpolation spaces. We then apply the results to the solution to a stochastic differential equation with a local H\"ormander condition or to the solution to the stochastic heat equation, in both cases under weak conditions on the coefficients relaxing the standard Lipschitz or H\"older continuity requests

Riesz transform and integration by parts formulas for random variables

We use integration by parts formulas to give estimates for the $L^p$ norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and Thalmaier. As a consequence, we obtain regularity and estimates for the density of non degenerated functionals on the Wiener space. We also give a semi-distance which characterizes the convergence to the boundary of the set of the strict positivity points for the density

On the distance between probability density functions

We give estimates of the distance between the densities of the laws of two functionals $F$ and $G$ on the Wiener space in terms of the Malliavin-Sobolev norm of $F-G.$ We actually consider a more general framework which allows one to treat with similar (Malliavin type) methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in $L^{1}$ of the densities

A hybrid approach for the implementation of the Heston model

We propose a hybrid tree-finite difference method in order to approximate the Heston model. We prove the convergence by embedding the procedure in a bivariate Markov chain and we study the convergence of European and American option prices. We finally provide numerical experiments that give accurate option prices in the Heston model, showing the reliability and the efficiency of the algorithm