Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

TL;DR

This paper develops a method to solve stochastic control problems with multiple default times by decomposing them into classical control problems under a reference filtration, simplifying analysis in default risk management.

Contribution

It introduces a decomposition technique for stochastic control problems under progressive filtration enlargement, extending control of diffusion processes with jumps.

Findings

Decomposition reduces complex control problems to classical ones.

Solutions expressed via BSDEs without jump terms.

Applications include defaultable claims and bilateral counterparty risk.

Abstract

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which are determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration.