Optimal Brownian stopping when the source and target are radially symmetric distributions

Abstract

Given two probability measures $\mu, \nu$ on $\mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $\tau$ that maximize/minimize the cost functional $\mathbb{E} |B_0 - B_\tau|^{\alpha}$, $\alpha > 0$, where $(B_t)_t$ is Brownian motion with initial law $\mu$ and with final distribution --once stopped at $\tau$-- equal to $\nu$. Under the assumption of radial symmetry on $\mu$ and $\nu$, we show that in dimension $d \geq 3$ and $\alpha \neq 2$, there exists a unique optimal solution given by a non-randomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem to the optimal transportation problem for subharmonic martingales, and establish a duality result. This paper is an expanded version of a previously posted but not published work by the authors.