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Forward Performance Processes under Multiple Default Risks

Wing Fung Chong, Roxana Dumitrescu, Gechun Liang, Kenneth Tsz Hin Ng

TL;DR

This work develops a forward-looking exponential utility for markets with multiple default risks by projecting the defaultable dynamics onto the default-free filtration through the Jacod-Pham decomposition. The authors construct a forward utility via a novel system of recursive infinite-horizon BSDEs with discounting, prove existence/uniqueness and boundedness of the solutions, and establish a verification framework linking these BSDEs to forward performance and optimal strategies. They extend the analysis to a stochastic-factor model with ergodicity, obtaining Markovian representations and uniform bounds, and define a long-run, risk-sensitive growth rate via ergodic BSDEs. The ergodic extension further clarifies how defaults shape the asymptotic performance and provides conditions under which the ergodic limit yields a well-defined forward utility, offering a robust toolkit for default contagion and long-term wealth optimization.

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.

Forward Performance Processes under Multiple Default Risks

TL;DR

This work develops a forward-looking exponential utility for markets with multiple default risks by projecting the defaultable dynamics onto the default-free filtration through the Jacod-Pham decomposition. The authors construct a forward utility via a novel system of recursive infinite-horizon BSDEs with discounting, prove existence/uniqueness and boundedness of the solutions, and establish a verification framework linking these BSDEs to forward performance and optimal strategies. They extend the analysis to a stochastic-factor model with ergodicity, obtaining Markovian representations and uniform bounds, and define a long-run, risk-sensitive growth rate via ergodic BSDEs. The ergodic extension further clarifies how defaults shape the asymptotic performance and provides conditions under which the ergodic limit yields a well-defined forward utility, offering a robust toolkit for default contagion and long-term wealth optimization.

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.
Paper Structure (29 sections, 17 theorems, 254 equations)

This paper contains 29 sections, 17 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Model Formulation
  3. Preliminaries
  4. Market Model
  5. Forward Utility Preference in Defaultable Markets
  6. Characterization of $\mathbb{G}$-Martingales under the Filtration $\mathbb{F}$
  7. $\mathbb{G}$-Forward Utility Preference and Characterizations under $\mathbb{F}$
  8. Exponential Forward Utility
  9. Recursive Infinite Horizon BSDEs
  10. Verification Theorem
  11. Stochastic Factor Market Model
  12. Infinite Horizon BSDEs and Markovian Solution
  13. Risk-Sensitive Growth Rate
  14. Construction of Forward Performance Processes via Ergodic BSDEs
  15. Concluding Remarks
  16. ...and 14 more sections

Key Result

Lemma 3.1

Let $M=(M_t)_{t\geq 0}$ be an $\mathbb{R}$-valued, $\mathbb{G}$-optional and integrable process (i.e., $\mathbb{E}[|M_t|]<\infty$ for any $t\geq 0$) with the following Jacod-Pham decomposition: where for any $n=0,\dots,m$, $M^n(\cdot,\cdot)\in \mathcal{O}_\mathbb{F}(\Delta_n,E^n;\mathbb{R})$. Suppose that $(M_t)_{t\geq 0}$ satisfies the following recursive relation: for any $n=0,\dots,m-1$, $(\th

Theorems & Definitions (44)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Definition 3.1
  • Theorem 3.1
  • Remark 4.1
  • ...and 34 more