Martingale representations in progressive enlargement by multivariate point processes

TL;DR

This paper establishes martingale representation results in the context of progressively enlarged filtrations driven by multivariate point processes, including conditions for strong representation and orthogonality of components.

Contribution

It extends martingale representation theory to multivariate point processes with conditions for strong representation and component orthogonality.

Findings

Weak representation up to explosion times.

Strong representation under infinite explosion time and discrete marks.

Orthogonality condition for martingale components.

Abstract

We show that all local martingales with respect to the initially enlarged natural filtration of a vector of multivariate point processes can be weakly represented up to the minimum among the explosion times of the components. We also prove that a strong representation holds if any multivariate point process of the vector has almost surely infinite explosion time and discrete mark's space. Then we provide a condition under which the components of the multidimensional local martingale driving the strong representation are pairwise orthogonal.