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An exit contract optimization problem

Xihao He, Xiaolu Tan, Jun Zou

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.

An exit contract optimization problem

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.

Paper Structure

This paper contains 15 sections, 13 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Exit contract optimization problem in continuous-time setting
  3. Mathematical formulation of the exit contract problem
  4. Agents' problem
  5. The exit contract optimization problem
  6. Solving the exit contract problem
  7. Further discussions and examples
  8. A discrete-time version and its convergence
  9. A discrete-time version of the exit contract problem
  10. Convergence of the discrete-time value function to the continuous-time one
  11. The problem with Markovian and/or continuous contract
  12. Conclusion
  13. Appendix
  14. The Snell envelop approach to the optimal stopping problem
  15. Bank-El Karoui's representation of stochastic processes

Key Result

Lemma 2.5

Let Assumption assum:f hold true. Then for every $L \in \mathcal{L}^+$, there exists an optional process $Y^L$ such that Moreover, $Y^L \in \mathcal{Y}$, and it has almost surely right-continuous paths.

Theorems & Definitions (34)

  • Definition 2.1: USCE
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.2
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 24 more