Optimal investment under multiple defaults risk: A BSDE-decomposition approach

TL;DR

This paper develops a decomposition method using backward stochastic differential equations to solve optimal investment problems in markets with multiple default risks, providing explicit solutions and numerical insights.

Contribution

It introduces a novel BSDE-decomposition approach for optimal investment under multiple defaults, enabling explicit characterization of solutions and strategies.

Findings

Existence and uniqueness of solutions to the BSDE system

Explicit characterization of value functions and strategies

Numerical tests illustrating default impact and asset correlation

Abstract

We study an optimal investment problem under contagion risk in a financial model subject to multiple jumps and defaults. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of default times is modeled by a conditional density hypothesis. In this Ito-jump process model, we give a decomposition of the corresponding stochastic control problem into stochastic control problems in the default-free filtration, which are determined in a backward induction. The dynamic programming method leads to a backward recursive system of quadratic backward stochastic differential equations (BSDEs) in Brownian filtration, and our main result proves, under fairly general conditions, the existence and uniqueness of a solution to this system, which characterizes explicitly the value function and optimal strategies to the optimal investment problem. We illustrate our solutions approach with some numerical tests emphasizing the impact of default intensities, loss or gain at defaults and correlation between assets. Beyond the financial problem, our decomposition approach provides a new perspective for solving quadratic BSDEs with a finite number of jumps.