Anticipative calculus with respect to filtered Poisson processes

Abstract

We construct the basis of a stochastic calculus for a new class of processes: filtered Poisson processes. These processes are defined by an fBm-like stochastic integral but a Poisson process is subsided to the Brownian motion. We use Malliavin calculus to first construct a gradient then a divergence operator, which will play the role of an anticipative stochastic integral. We study into details the sample-paths regularity of this integral and give an Itô formula for Itô-like processes.  2005 Elsevier SAS. All rights reserved. RésuméLes processus de Poisson filtrés sont définis par une intégrale stochastique d'un noyau déterministe relativement à un processus de Poisson. Ils sont au processus de Poisson ce que le mouvement brownien fractionnaire est au mouvement brownien ordinaire. Nous construisons pour ces processus les bases du calcul anticipatif. Ce travail similaire à celui d'Oksendal et ses collaborateurs sur les processus de sauts de Lévy, emprunte une autre approche que celle du calcul de Hida. Nous introduisons l'opérateur gradient sans utiliser la décomposition en chaos puis nous définissons son adjoint, ce qui nous permet de construire une notion d'intégrale anticipante. Les propriétés de régularité trajectorielle de cette intégrale sont précisées. Nous terminons notre travail par une formule d'Itô pour ces processus.

Quasi-invariance and integration by parts for determinantal and permanental processes

“…We here address the problem within the point of view of Malliavin calculus. To date, Malliavin calculus for point processes has been developed namely for Poisson processes [2,6,7,10,13,22] and some of their extensions: Gibbs processes [3], marked processes [4], filtered Poisson processes [13], cluster processes [9] and Lévy processes [5,14]. There exist three approaches to construct a Malliavin calculus framework for point processes: one based on white noise analysis, one based on a difference operator and chaos decomposition and one which relies on quasi-invariance of the law of Poisson process with respect to some perturbations.…”

Functionals of a Lévy Process on Canonical and Generic Probability Spaces

“…Splitting up the last term into two summands, using (7) to get F X t + v1I [0,t] (r) t≥0 = Y + vD r,v Y, P ⊗ Ñ-a.e., in L 0 (Ω, F , P) and L 0 (P ⊗ Ñ), respectively. Equation (8) implies that the definition of G (X, Y + vD r,v Y ) is meaningful and does not depend on the choice of the functional G up to functions being zero P ⊗ Ñ-a.e.…”

“…Since we have by Lemma 3.1 that Y = F (X), P-a.s., we can use equation (8) to get f (·, Y ) = G(X, F (X)), P-a.s.…”

“…The other way is to consider "true" derivative operators, where a chain rule is possible. Those are given for example by Carlen and Pardoux [6], Denis [10], Decreusefond and Savy [8], Bally et al [2] and Kulik [19].…”

Time Change, Volatility, and Turbulence

“…Stochastic integration in these general settings is discussed for BVP in [Hu03], [Dec05], [DecSa06], cf. also [NoVaVir99], and for LVP in [BeMar07].…”