Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.     

Invariant or quasi-invariant probability measures for infinite dimensional groups

Deterministic Euler flow on a torus cannot leave invariant any probability measure.

To the Japanese Mathematical Community, with my admiration and my warmest friendship.

This article is based on the 3rd Takagi Lectures that the author delivered at Graduate School of Mathematical Sciences, the University of Tokyo on November 23, 2007.     

Representations for conditional expectations and applications to pricing and hedging of financial products in Lévy and jump-diffusion setting

In this article, we derive expressions for conditional expectations in terms of regular expectations without conditioning but involving some weights. For this purpose, we apply two approaches: the conditional density method and the Malliavin method. We use these expressions for the numerical estimation of the price of American options and their deltas in a Lévy and jump-diffusion setting. Several examples of applications to financial and energy markets are given including numerical examples.     

Parameter estimation for stochastic equations with additive fractional Brownian sheet

We study the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet. We use the Girsanov transform for the the two-parameter fractional Brownian motion, as well as the Malliavin calculus and Gaussian regularity theory.      

The law of the solution to a nonlinear hyperbolicSPDE

Let { _s,t ,(s,t ₊ ² } be a white noise on ₊ ² . We consider the hyperbolic stochastic partial differential equation {ie863-3} The purpose of this paper is to study the law of the solution to this equation. We analyze the existence and smoothness of the density using the tools of Malliavin Calculus. Finally we prove a large deviation principle on the space of continuous functions, for the family of probabilities obtained by perturbation of the noise in the equation.This work has been partially supported by the grant of the DGICYT No. PB 930052 and the EU Science project CT 910459.     

D∞-APPROXIMATION OF PRODUCT VARIATIONS OF TWO PARAMETER SMOOTH SEMIMARTINGALES

Let X be a two parameter smooth semimartingale and (~X) be its process of the product variation. It is proved that (~X) can be approximated as D∞-limit of sums of its discrete product variations as the mesh of division tends to zero. Moreover, this result can be strengthen to yield the quasi sure convergence of sums by estimating the speed of the convergence.     

On Strongly Petrovskii's Parabolic SPDEs in Arbitrary Dimension and Application to the Stochastic Cahn–Hilliard Equation

In this paper we show that the Cahn–Hilliard stochastic PDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. We study the regularity of the trajectories of the solution and the absolute continuity of its law at some given time and position. This is done by showing a priori estimates which heavily depend on the specific equation, and by proving general results on stochastic and deterministic integrals involving general operators on smooth domains of ^d which are parabolic in the sense of Petrovskii, and do not necessarily define a semi-group of operators. These last estimates might be used in a more general framework.     

Logarithmic Estimates for the Density of Hypoelliptic Two-Parameter Diffusions

We consider a hypoelliptic two-parameter diffusion. We first prove a sharp upper bound in small time (s, t)[0, 1]² for the L^p-moments of the inverse of the Malliavin matrix of the diffusion process. Second, we establish the behaviour of2² log p_s, t(x, y), as ↓0, where x is the initial condition of the diffusion, = , and p_s, t(x, y) is the density of the hypoelliptic two-parameter diffusion.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾     

Densities of One-Dimensional Backward SDEs

This work is devoted to the study of the existence and smoothness of the marginal densities of the solution of one-dimensional backward stochastic differential equations. Under monotonicity conditions of a function of the coefficients, we obtain that the smoothness properties of the forward process influencing the backward equation, transfer to the densities of the solution. Once established these conditions, we apply the result to study the tail behavior of the solution process. Mathematics Subject Classification (2000) 60H10.Fabio Antonelli: The first author was partially supported by the MIUR COFIN grant 2000.Arturo Kohatsu-Higa: The second author was partially supported by grants BFM2003-03324 and BFM 2003-04294. The authors wish to thank the referee for his/her comments.