On The Weak Representation Property in Progressively Enlarged Filtrations with an Application to Exponential Utility Maximization

TL;DR

This paper demonstrates that the weak representation property of semimartingales is maintained when enlarging filtrations with a specific type of random time, enabling solutions to utility maximization problems via BSDEs.

Contribution

It establishes the preservation of the weak representation property under progressive enlargement by certain random times, and applies this to solve utility maximization problems using BSDEs.

Findings

Weak representation property is preserved under filtration enlargement.

Enables solving utility maximization problems in enlarged filtrations.

Application of BSDEs to dynamic utility maximization over different time horizons.

Abstract

In this paper we show that the weak representation property of a semimartingale $X$ with respect to a filtration $\mathbb{F}$ is preserved in the progressive enlargement $\mathbb{G}$ by a random time $\tau$ avoiding $\mathbb{F}$-stopping times and such that $\mathbb{F}$ is immersed in $\mathbb{G}$. As an application of this, we can solve an exponential utility maximization problem in the enlarged filtration $\mathbb{G}$ following the dynamical approach, based on suitable BSDEs, both over the fixed time horizon $[0,T]$, $T>0$, and over $[0,T\wedge\tau]$.