On the diameter of the stopped spider process

TL;DR

This paper determines the optimal constant for an inequality involving the diameter of Walsh Brownian motion (spider process) and its expected stopping time, using an explicit optimal stopping problem solution.

Contribution

It identifies the best constant in a diameter inequality for Walsh Brownian motion, advancing understanding of its stopping time properties.

Findings

Established the optimal constant C_n for the diameter inequality.

Explicitly solved the associated optimal stopping problem.

Provided bounds and characterization of the process's diameter behavior.

Abstract

We consider the Brownian ``spider process'', also known as Walsh Brownian motion, first introduced in the epilogue of Walsh 1978. The paper provides the best constant $C_n$ for the inequality $$ E D_\tau\leq C_n \sqrt{E \tau},$$ where $\tau$ is the class of all adapted and integrable stopping times and $D$ denotes the diameter of the spider process measured in terms of the British rail metric. The proof relies on the explicit identification of the value function for the associated optimal stopping problem.