Josep Lluís Solé

Stein’s method and Normal approximation of Poisson functionals

We combine Steins method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Esseen bounds in Central limit theorems (CLTs) involving multiple Wiener-Ito integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck Levy processes.

On Lévy processes, Malliavin calculus and market models with jumps

Abstract. Recent work by Nualart and Schoutens (2000), where a kind of chaotic property for Lévy processes has been proved, has enabled us to develop a Malliavin calculus for Lévy processes. For simple Lévy processes some useful formulas for computing Malliavin derivatives are deduced. Applications for option hedging in a jump–diffusion model are given.

Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

ABSTRACT We give the Wiener–Itoˆ chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev–Watanabe spaces.

Chaos Expansions and Malliavin Calculus for Lévy Processes

There are two different chaos expansions of a square integrable functional of a Lévy process: one proved by Itô [9] and the other by Nualart and Schoutens [17]. Related to each expansion a Malliavin type Calculus has been developed, being both quite different. In this paper we review the relationship between both approaches, and compare the corresponding Clark–Ocone–Haussmann representation formula.

Time-space harmonic polynomials relative to a Levy process

In this work, we give a closed form and a recurrence relation for a family of time-space harmonic poly nomials relative to a Levy process. We also state the relationship with the Kailath-Segall (orthogonal) polynomials associated to the process.

Chaotic Kabanov formula for the Azema martingales

We derive the chaotic expansion of the product of nth- and first-order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulae for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties, relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes.

A pathwise approach to backward and forward stochastic differential equations on the poisson space

We study the existence and uniqueness of pathwise solutions to backward and forward stochastic differential equations on the Poisson space. We obtain the structure of these pathwise solutions to give the relationship between them. Also, in the bilinear case, we calculate the explicit form of their chaos decompositions. *Partially supported by DGICYT grant PB93-0052, PB96-1182 and CIRIT grant 97-SGR00144.

Malliavin Calculus for Stochastic Processes and Random Measures with Independent Increments

Malliavin calculus for Poisson processes based on the difference operator or add-one-cost operator is extended to stochastic processes and random measures with independent increments. Our approach is to use a Wiener–Ito chaos expansion, valid for both stochastic processes and random measures with independent increments, to construct a Malliavin derivative and a Skorohod integral. Useful derivation rules for smooth functionals given by Geiss and Laukkarinen (Probab Math Stat 31:1–15, 2011) are proved. In addition, characterizations for processes or random measures with independent increments based on the duality between the Malliavin derivative and the Skorohod integral following an interesting point of view from Murr (Stoch Process Appl 123:1729–1749, 2013) are studied.

Local Malliavin calculus for Lévy processes and applications

The Malliavin derivative operator for the Poisson process introduced by Carlen and Pardoux [Differential calculus and integration by parts on a Poisson space, in Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, S. Albeverio et al. (eds), Kluwer, Dordrecht, 1990, pp. 63–73] is extended to Lévy processes. It is a true derivative operator (in the sense that it satisfies the chain rule), and we deduce a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyse the absolute continuity of the law of the solution of some stochastic differential equations with jumps.