Distribution-constrained maximum stopping of maximum type

2026-10-01

AI Summary

This study addresses the stochastic optimal stopping problem for the maximum of Brownian motion under distributional constraints, focusing on characterizing its optimal boundary. To this end, it proposes a novel variational inequality formulation based on a time-reversed probabilistic representation. By combining the construction of discrete Snell envelopes with a viscosity solution comparison principle, the work overcomes the challenge of compatibility estimates at singular corners within the space-time domain. The main contributions include rigorous proofs of the existence and uniqueness of the optimal stopping time, as well as establishing its optimality as the hitting time of a specific boundary. These results substantially refine and complete the theoretical framework in this area of research.

Abstract

We consider the distribution-constrained optimal stopping problem $\sup_{τ\sim μ} \mathbb E[B^*_τ]$, where $μ$ is a probability distribution on $\mathbb R_+$, and $(B^*_t)$ denotes the running maximum of a standard Brownian motion. This problem was introduced in Beiglbock et al. (PTRF, 2018), where a monotonicity principle is used to establish the optimal stopping time as the hitting time of a specific boundary. In this paper, we characterize this boundary by a variational inequality. In the spirit of Cox et al. (PTRF, 2019), we provide a novel probabilistic representation for the variational inequality as a time-reversed optimal stopping problem. A key ingredient for proving the viscosity solution property and comparison principle is a quantitative estimate near the singular corner of the time-space domain, where the initial and boundary conditions are incompatible. We then prove the optimality of the resulting hitting time through a discrete-time Snell envelope construction and a stability argument for the associated stopping times.

Problem

• distribution-constrained optimal stopping
• Brownian motion running maximum
• variational inequality
• hitting time boundary

Innovation

• distribution-constrained optimal stopping
• variational inequality
• time-reversed optimal stopping
• viscosity solution
• Snell envelope