Chaotic Kabanov Formula for the Azéma Martingales

Abstract

We derive the chaotic expansion of the product of nth-and ®rst-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.

The chaotic-representation property for a class of normal martingales

“…When F 0 is trivial, the following result may be derived from the chaotic Kabanov formula of Privault et al [15,Theorem 1].…”

Anticipating stratonovich integral with respect to the Azema's martingales

“…After some works (see for example [8] or [12] ) we have several tools in the context of normal martingales. One of these tools is the definition of an anticipating integral (see [8] ) and the other is the chaotic Kabanov formula (see [12] ) obtained for a certain class of normal martingales.…”

Normal martingales and polynomial families

“…The problem studied in this paper is to find if there exist other normal martingales which are associated to classical sequences of polynomials. Privault, Solé and Vives [5] solved this problem via the quantum Kabanov formula under some assumptions on the normal martingales considered. We solve the problem without these assumptions and we give a complete study of this subject in Section 2.…”

“…As we mentioned in the abstract, Privault, Solé and Vives [5] studied the question of whether there exist other normal martingales X whose iterated integrals P (n) can be expressed as polynomials in X according to the following definition: Definition 1. We will say that a normal martingale X has an associated family of polynomials (Q…”

“…They proved, by applying the quantum Kabanov formula [5], that only Wiener and compensated Poisson processes can satisfy Definition 1.…”