Enlargement of filtration on Poisson space: a Malliavin calculus approach

Abstract

Being inspired by the work of Imkeller et al., who first developed a measured-valued version of Malliavin calculus to study the enlargement of filtration by an ‘exogenous’ random variable on the Wiener space, in this article we consider the enlargement of filtration on the Poisson space. We construct a measure-valued Malliavin calculus on the Poisson space with the Malliavin type calculus defined by Mensi and Privault. Without resorting to the traditional Jacod’s condition nor its local version, the measure-valued Poisson Malliavin calculus can be applied to compute the ‘information drift’. Our result applies for a general -integrable ‘exogenous’ random variable, while previous works require a more restrictive Malliavin differentiability. Finally, we illustrate a concrete application of our results with an example from Mensi and Privault.