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Grab It Before It's Gone: Testing Uncertain Rewards under a Stochastic Deadline

Steven Campbell, Georgy Gaitsgori, Richard Groenewald, Ioannis Karatzas

TL;DR

The paper addresses optimal stopping for an unknown two-point drift observed through $X(t)=R t+W(t)$ under a stochastic deadline $\gamma$, formulating the problem in a Bayesian filtering framework with a posterior probability $\Pi(t)$. It develops a comprehensive theory: reformulating the problem as a Markovian optimal stopping problem, connecting it to American options, and proving existence, smooth-fit, and a free-boundary PDE in the continuation region, along with an integral equation for the boundary. The authors obtain regularity results, boundary continuity under structural assumptions, and provide numerical illustrations across diverse horizon-distribution scenarios. The work advances sequential analysis with random horizons by unifying drift-detection, optimal stopping, and option-theoretic methods, yielding a robust framework for stochastic-deadline problems with practical implications for sequential hiring, testing, and decision-making under uncertainty.

Abstract

We study a sequential estimation problem for an unknown reward in the presence of a random horizon. The reward takes one of two predetermined values which can be inferred from the drift of a Wiener process, which serves as a signal. The objective is to use the information in the signal to estimate the reward which is made available until a stochastic deadline that depends on its value. The observer must therefore work quickly to determine if the reward is favorable and claim it before the deadline passes. Under general assumptions on the stochastic deadline, we provide a full characterization of the solution that includes an identification with the unique solution to a free-boundary problem. Our analysis derives regularity properties of the solution that imply its ``smooth fit'' with the boundary data, and we show that the free-boundary solves a particular integral equation. The continuity of the free-boundary is also established under additional structural assumptions that lead to its representation in terms of a continuous transformation of a monotone function. We provide illustrations for several examples of interest.

Grab It Before It's Gone: Testing Uncertain Rewards under a Stochastic Deadline

TL;DR

The paper addresses optimal stopping for an unknown two-point drift observed through under a stochastic deadline , formulating the problem in a Bayesian filtering framework with a posterior probability . It develops a comprehensive theory: reformulating the problem as a Markovian optimal stopping problem, connecting it to American options, and proving existence, smooth-fit, and a free-boundary PDE in the continuation region, along with an integral equation for the boundary. The authors obtain regularity results, boundary continuity under structural assumptions, and provide numerical illustrations across diverse horizon-distribution scenarios. The work advances sequential analysis with random horizons by unifying drift-detection, optimal stopping, and option-theoretic methods, yielding a robust framework for stochastic-deadline problems with practical implications for sequential hiring, testing, and decision-making under uncertainty.

Abstract

We study a sequential estimation problem for an unknown reward in the presence of a random horizon. The reward takes one of two predetermined values which can be inferred from the drift of a Wiener process, which serves as a signal. The objective is to use the information in the signal to estimate the reward which is made available until a stochastic deadline that depends on its value. The observer must therefore work quickly to determine if the reward is favorable and claim it before the deadline passes. Under general assumptions on the stochastic deadline, we provide a full characterization of the solution that includes an identification with the unique solution to a free-boundary problem. Our analysis derives regularity properties of the solution that imply its ``smooth fit'' with the boundary data, and we show that the free-boundary solves a particular integral equation. The continuity of the free-boundary is also established under additional structural assumptions that lead to its representation in terms of a continuous transformation of a monotone function. We provide illustrations for several examples of interest.

Paper Structure

This paper contains 26 sections, 22 theorems, 162 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Generalized model
  4. Equivalent Formulations
  5. A Posteriori Probability Process and General Markovian Setup
  6. General Markovian Framework
  7. Reverting Back to the Original Problem
  8. An Equivalent Formulation via a Different Payoff Function
  9. An Equivalent Formulation via American Options
  10. Analysis of the Optimal Stopping Problem
  11. Preliminaries and the Main Results
  12. Properties of the Value Function
  13. Study of the Stopping Boundary
  14. Properties of the Boundary
  15. Sufficient Conditions for Monotonicity of the Boundary under an Appropriate Transformation
  16. ...and 11 more sections

Key Result

Lemma 3.1

The conditional probability process $\Pi(\cdot)$, defined in conditional_probability_process_def, satisfies where the process known as the "innovation process", is an $\mathbb{F}$--Brownian motion. The processes $B(\cdot)$, $X(\cdot)$, and $\Pi(\cdot)$ generate the same filtration $\mathbb{F}$, and the process $\Pi(\cdot)$ is a strong Markov process with respect to this filtration. Furthermore,

Figures (2)

  • Figure 5.1: Optimal stopping boundary for the parameters and discount functions of Examples \ref{['eg1']}, \ref{['eg2']}, and \ref{['eg3']}, (left), and \ref{['eg4']} (right).
  • Figure 5.2: Optimal stopping boundary for the parameters and discount functions of Examples \ref{['eg5']} (left) and \ref{['eg6']} (right).

Theorems & Definitions (54)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • ...and 44 more