# Martingale Representation in the Enlargement of the Filtration Generated by a Point Process

## TL;DR

This paper investigates martingale representation theorems within an enlarged filtration generated by a point process, extending existing results to cases involving initial and progressive enlargements without additional assumptions.

## Contribution

It provides a general martingale representation theorem for filtrations enlarged by a point process, including the special case of enlargement by a random time.

## Key findings

- Derived martingale representation theorems for enlarged filtrations
- No assumptions needed on the dependence between processes
- Includes the case of enlargement by a random time

## Abstract

Let $X$ be a point process and let $\mathbb{X}$ denote the filtration generated by $X$. In this paper we study martingale representation theorems in the filtration $\mathbb{G}$ obtained as an initial and progressive enlargement of the filtration $\mathbb{X}$. The progressive enlargement is done here by means of a whole point process $H$. We do not require further assumptions on the point process $H$ nor on the dependence between $X$ and $H$. In particular, we recover the special case of the progressive enlargement by a random time $\tau$.