On controller-stopper problems with jumps and their applications to indifference pricing of American options

TL;DR

This paper develops a decomposition approach for controller-stopper problems with jumps, applying it to indifference pricing of American options under default risk, resulting in a system of reflected backward stochastic differential equations.

Contribution

It introduces a novel decomposition method for jump-including controller-stopper problems, enabling analysis via RBSDEs for American option pricing under default risk.

Findings

Decomposition reduces complex jump problems to Brownian filtration problems.

Existence of solutions to the RBSDE system is established.

The solution characterizes the value function for the problem.

Abstract

We consider controller-stopper problems in which the controlled processes can have jumps. The global filtration is represented by the Brownian filtration, enlarged by the filtration generated by the jump process. We assume that there exists a conditional probability density function for the jump times and marks given the filtration of the Brownian motion and decompose the global controller-stopper problem into controller-stopper problems with respect to the Brownian filtration, which are determined by a backward induction. We apply our decomposition method to indifference pricing of American options under multiple default risk. The backward induction leads to a system of reflected backward stochastic differential equations (RBSDEs). We show that there exists a solution to this RBSDE system and that the solution provides a characterization of the value function.