BSDEs in Utility Maximization with BMO Market Price of Risk

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.

33 pages, 1 figure

References in corpus (8)

• Utility maximization in incomplete markets
• BSDE with quadratic growth and unbounded terminal value
• Quadratic BSDEs with convex generators and unbounded terminal conditions
• Dynamic exponential utility indifference valuation
• Bounded solutions to backward SDE's with jumps for utility optimization and indifference hedging
• The Opportunity Process for Optimal Consumption and Investment with Power Utility
• Risk Aversion Asymptotics for Power Utility Maximization
• The Bellman equation for power utility maximization with semimartingales

Cited by in corpus (2)

• New proofs of some results on BMO martingales using BSDEs
• On Quadratic BSDEs with Final Condition in L2