The Cayley-Moser problem with Poissonian arrival of offers

TL;DR

This paper analyzes a continuous-time optimal stopping problem where offers arrive randomly over time, providing explicit solutions for the optimal selling policy, sale price distribution, and stopping time, with applications to various offer distributions.

Contribution

It introduces a tractable continuous-time formulation of the Cayley-Moser problem with Poisson arrivals, deriving explicit solutions and distributions for the optimal policy.

Findings

Explicit differential equation solution for optimal policy

Distribution formulas for sale price and stopping time

Application to multiple offer distributions

Abstract

We study a version of the classical Cayley-Moser optimal stopping problem, in which a seller must sell an asset by a given deadline, with the offers, which are independent random variables with a known distribution, arriving at random times, as a Poisson process. This continuous-time formulation of the problem is much more analytically tractable than the analogous discrete-time problem which is usually considered, leading to a simple differential equation that can be explicitly solved to find the optimal policy. We study the performance of this optimal policy, and obtain explicit expressions for the distribution of the realized sale price, as well as for the distribution of the stopping time. The general results are used to explore characteristics of the optimal policy and of the resulting bidding process, and are illustrated by application to several specific instances of the offer distribution.