An exit contract optimization problem

TL;DR

This paper addresses the complex problem of designing a universal exit contract for multiple heterogeneous agents, transforming it into an optimal control problem and providing methods for approximation and optimization.

Contribution

It introduces a novel approach to exit contract design using stochastic control and multiple stopping frameworks, with proofs of existence and approximation methods.

Findings

Existence of an optimal universal exit contract proved.

Reduction of the problem to an optimal control and multiple stopping problem.

Development of a discrete-time approximation method for continuous-time problems.

Abstract

We study an exit contract design problem, where one provides a universal exit contract to multiple heterogeneous agents, with which each agent chooses an optimal (exit) stopping time. The problem consists in optimizing the universal exit contract w.r.t. some criterion depending on the contract as well as the agents' exit times. Under a technical monotonicity condition, and by using Bank-El Karoui's representation of stochastic processes, we are able to transform the initial contract optimization problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is proved. We next show that the problem in the continuous-time setting can be approximated by a sequence of discrete-time ones, which would induce a natural numerical approximation method. We finally discuss the optimaization problem over the class of all Markovian and/or continuous exit contracts.