On the Diameter of the Stopped Spider Process

Abstract

We consider the Brownian “spider process,” also known as Walsh Brownian motion, first introduced by J. B. Walsh [Walsh JB (1978) A diffusion with a discontinuous local time. Asterisque 52:37–45]. The paper provides the best constant Cn for the inequality[Formula: see text]where τ 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. This solves a variant of the long-standing open “spider problem” due to L. E. Dubins. The proof relies on the explicit identification of the value function for the associated optimal stopping problem. Funding: P. A. Ernst thanks the Royal Society Wolfson Fellowship (RSWF\R2\222005) and the U.S. Office of Naval Research (ONR N00014-21-1-2672) for their support of this research.