Martingale representation on enlarged filtrations: the role of the accessible jump times

TL;DR

This paper investigates how accessible jump times influence the martingale representation in enlarged filtrations, providing new insights and a novel theorem for representing martingales in this context.

Contribution

It introduces a detailed analysis of accessible jump times' impact on martingale representation and establishes a new martingale representation theorem for enlarged filtrations.

Findings

Accessible jump times affect Jacod's dimension of martingale spaces.

New martingale representation theorem on enlarged filtrations.

Analysis of overlapping jump times in different filtrations.

Abstract

We consider a filtration $\mathbb{G}$ obtained as enlargement of a filtration $\mathbb{F}$ by a filtration $\mathbb{H}$. We assume that all $\mathbb{F}$-local martingales are represented by a martingale $M$ and all $\mathbb{H}$-local martingales are represented by a martingale $N$. $M$ and $N$ are not necessarily quasi-left continuous processes and their jump times may overlap. We first analyze the contribution of the accessible jump times of $M$ and $N$ to the Jacod's dimension of the space of the $\mathcal{H}^1(\mathbb{G})$-martingales. Then we prove a new martingale representation theorem on $\mathbb{G}$.