Paper: arXiv 2610.01585

Authors: Shuoqing Deng, Xin Zhang

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly mathematical and theoretical approach to an optimal stopping problem, focusing on variational inequalities and probabilistic representations. While the mathematical framework is sophisticated and novel, there is no empirical validation or backtesting, which is typical for theoretical probability papers.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize Optimal Stopping Boundary for Distribution-Constrained Max Stopping] --> B{Key Methodology: Variational Inequality & Probabilistic Representation};
    B --> C{Inputs: Standard Brownian Motion (B*t), Distribution μ on R+};
    C --> D[Computational Processes: Time-reversed Optimal Stopping, Viscosity Solution, Snell Envelope];
    D --> E[Key Findings: Optimal Boundary Characterized by VI, Novel Probabilistic Representation, Optimality Proof via Snell Envelope];