BSDEs in Utility Maximization with BMO Market Price of Risk

TL;DR

This paper investigates quadratic BSDEs in power utility maximization with BMO market price of risk, establishing conditions for solutions and highlighting non-uniqueness issues in a Brownian framework.

Contribution

It provides a necessary and sufficient condition for the existence of solutions and introduces a new minimal exponential moments condition for bounded solutions.

Findings

Existence of solutions characterized by a new condition.

Non-uniqueness of solutions due to stochastic exponential representation.

Examples illustrating the sharpness of theoretical results.

Abstract

This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum of distinct square-integrable solutions. This feature occurs since, contrary to the classical Ito representation theorem, a representation of random variables in terms of stochastic exponentials is not unique. We study in detail when the BSDE has a bounded solution and derive a new dynamic exponential moments condition which is shown to be the minimal sufficient condition in a general filtration. The main results are complemented by several interesting examples which illustrate their sharpness as well as important properties of the utility maximization BSDE.