OPTIMAL INVESTMENT UNDER MULTIPLE DEFAULTS RISK: A BSDE-DECOMPOSITION APPROACH

• Published in: The Annals of applied probability Vol. 23; no. 2; pp. 455 - 491
• Main Authors: Jiao, Ying, Kharroubi, Idris, Pham, Huyên
• Format: Journal Article
• Language: English
• Published: Hayward Institute of Mathematical Statistics 01-04-2013; Institute of Mathematical Statistics (IMS); The Institute of Mathematical Statistics
• Subjects: 60J75; 91B28; 93E20; Credit risk; Default; Density; Differential equations; Dynamic programming; Financial investments; Financial portfolios; Investment policy; Investment portfolios; Investment risk; Investment strategies; Martingales; Mathematical problems; Mathematics; multiple defaults; Optimal investment; Optimal strategies; Probability; progressive enlargement of filtrations; quadratic backward stochastic differential equations; Risk management; Stochastic control theory; Utility functions
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/11-AAP829

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 Itô-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.