Forward Performance Processes under Multiple Default Risks

2026-01-05

AI Summary

This study addresses dynamic investment performance evaluation under multiple default risks. Building upon the Jacod–Pham decomposition, the authors construct a forward exponential utility within the default-free information flow and characterize its dynamic structure via an infinite-horizon discounted recursive backward stochastic differential equation (BSDE) system. They extend forward performance theory to a multi-default setting for the first time, establishing supermartingale and martingale criteria linked to the default-free filtration. In a model featuring ergodic stochastic factors, they prove existence, uniqueness, and uniform boundedness of the BSDE solution, derive uniform estimates for its Markovian representation, and uncover a correspondence between the constant term in the long-term limiting BSDE and the risk-sensitive growth rate of the optimal wealth process.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Multiagent Systems: Multiagent Systems under Uncertainty
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Forward Performance Processes
• Multiple Default Risks
• Backward Stochastic Differential Equations
• Martingale Property
• Ergodic Dynamics

Innovation

• forward performance processes
• multiple default risks
• backward stochastic differential equations (BSDEs)
• Jacod-Pham decomposition
• ergodic limit