Forward Performance Processes under Multiple Default Risks

TL;DR

This paper develops a forward exponential utility framework in markets with multiple default risks, using advanced stochastic calculus and BSDEs, and extends the analysis to ergodic stochastic factor models to understand long-term growth rates.

Contribution

It introduces a novel construction of forward performance processes in defaultable markets using recursive BSDEs and extends the theory to ergodic models with uniform bounds and long-term growth insights.

Findings

Established existence, uniqueness, and boundedness of solutions to the BSDE system.

Characterized the supermartingale property of the performance process.

Derived the risk-sensitive long-run growth rate in the ergodic limit.

Abstract

This article constructs a forward exponential utility in a market with multiple defaultable risks. Using the Jacod-Pham decomposition for random fields, we first characterize forward performance processes in a defaultable market under the default-free filtration. We then construct a forward utility via a system of recursively defined, indexed infinite-horizon backward stochastic differential equations (BSDEs) with discounting, and establish the existence, uniqueness, and boundedness of their solutions. To verify the required (super)martingale property of the performance process, we develop a rigorous characterization of this property with respect to the general filtration in terms of a set of (in)equalities relative to the default-free filtration. We further extend the analysis to a stochastic factor model with ergodic dynamics. In this setting, we derive uniform bounds for the Markovian solutions of the infinite-horizon BSDEs, overcoming technical challenges arising from the special structure of the system of BSDEs in the defaultable setting. Passing to the ergodic limit, we identify the limiting BSDE and relate its constant to the risk-sensitive long-run growth rate of the optimal wealth process.