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The non-linear multiple stopping problem: between the discrete and the continuous time

Miryana Grigorova, Marie-Claire Quenez, Peng Yuan

TL;DR

The paper develops a general theory for non-linear optimal multiple stopping with evaluation operators $\rho_{S,\tau}$ that depend on both the evaluation time and horizon, using Bermudan stopping strategies to bridge discrete and continuous time. It introduces a reduction mechanism for the double stopping problem and extends it to a nonlinear $d$-stopping framework, providing necessary and sufficient optimality conditions, existence results, and constructive schemes for optimal stopping times, including symmetric reward cases. Concrete examples are given via nonlinear $g$-evaluations from BSDEs and dynamic concave utilities, illustrating horizon-risk modeling and non-linear evaluation in finance-like settings. The results yield a versatile, horizon-aware SNell-envelope approach for non-linear evaluations and swing-like options, with potential numerical implications for complex stopping problems under general risk measures.

Abstract

We consider the non-linear optimal multiple stopping problem under general conditions on the non-linear evaluation operators, which might depend on two time indices: the time of evaluation/assessment and the horizon (when the reward or loss is incurred). We do not assume convexity/concavity or cash-invariance. We focus on the case where the agent's stopping strategies are what we call Bermudan stopping strategies, a framework which can be seen as lying between the discrete and the continuous time. We first study the non-linear double optimal stopping problem by using a reduction approach. We provide a necessary and a sufficient condition for optimal pairs, and a result on existence of optimal pairs. We then generalize the results to the non-linear $d$-optimal stopping problem. We treat the symmetric case (of additive and multiplicative reward families) as examples.

The non-linear multiple stopping problem: between the discrete and the continuous time

TL;DR

The paper develops a general theory for non-linear optimal multiple stopping with evaluation operators that depend on both the evaluation time and horizon, using Bermudan stopping strategies to bridge discrete and continuous time. It introduces a reduction mechanism for the double stopping problem and extends it to a nonlinear -stopping framework, providing necessary and sufficient optimality conditions, existence results, and constructive schemes for optimal stopping times, including symmetric reward cases. Concrete examples are given via nonlinear -evaluations from BSDEs and dynamic concave utilities, illustrating horizon-risk modeling and non-linear evaluation in finance-like settings. The results yield a versatile, horizon-aware SNell-envelope approach for non-linear evaluations and swing-like options, with potential numerical implications for complex stopping problems under general risk measures.

Abstract

We consider the non-linear optimal multiple stopping problem under general conditions on the non-linear evaluation operators, which might depend on two time indices: the time of evaluation/assessment and the horizon (when the reward or loss is incurred). We do not assume convexity/concavity or cash-invariance. We focus on the case where the agent's stopping strategies are what we call Bermudan stopping strategies, a framework which can be seen as lying between the discrete and the continuous time. We first study the non-linear double optimal stopping problem by using a reduction approach. We provide a necessary and a sufficient condition for optimal pairs, and a result on existence of optimal pairs. We then generalize the results to the non-linear -optimal stopping problem. We treat the symmetric case (of additive and multiplicative reward families) as examples.

Paper Structure

This paper contains 14 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The framework
  3. The optimal non-linear double stopping problem
  4. Reduction approach
  5. Existence of an optimal pair $(\tau_{1}^{*}, \tau_{2}^{*})$ for $V(S)$
  6. Non-linear optimal $d$-stopping problem
  7. Reduction approach
  8. The particular symmetric case
  9. Some particular examples of symmetric rewards
  10. Optimal stopping times in the non-linear optimal $d$-stopping problem
  11. Examples of non-linear operators
  12. Non-linear $g$-evaluations induced by BSDEs
  13. Dynamic concave utilities
  14. Appendix

Key Result

Proposition 1

i) $(V(S))_{S \in \Theta}$ is an admissible family. [0.2cm] ii) (maximising sequence property) There exists a sequence of pairs of Bermudan stopping times $(\tau_{1}^{n}, \tau_{2}^{n}) \in \Theta_{S} \times \Theta_{S}$ such that iii) The value family $(V(S))_{S \in \Theta}$ is a $(\Theta, \rho)$ - supermartingale family.

Theorems & Definitions (23)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Proposition 1
  • Theorem 1: Reduction
  • Proposition 2: Construction of optimal stopping times for $V(S)$. Sufficient condition
  • Proposition 3: A necessary condition for optimality
  • Theorem 2: Existence of an optimal pair
  • ...and 13 more