Distribution-constrained maximum stopping of maximum type
Abstract
We consider the distribution-constrained optimal stopping problem supτ∼ μ 𝔼[B*τ], where μ is a probability distribution on 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.
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BibTeX
@misc{deng2026distributionconstrained,
title = {{Distribution-constrained maximum stopping of maximum type}},
author = {Shuoqing Deng and Xin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01585},
eprint = {2610.01585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01585},
}
RIS
TY - UNPB
TI - Distribution-constrained maximum stopping of maximum type
AU - Shuoqing Deng
AU - Xin Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01585
UR - https://arxiv.org/abs/2610.01585
AB - We consider the distribution-constrained optimal stopping problem sup_τ∼ μ 𝔼[B^*_τ], where μ is a probability distribution on 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.
ER -