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

## TL;DR

This paper characterizes optimal Brownian stopping times between radially symmetric distributions in higher dimensions, linking the problem to subharmonic martingale transportation and establishing duality results.

## Contribution

It provides a unique, explicit description of optimal stopping times as hitting times to symmetric barriers under radial symmetry assumptions.

## Key findings

- Existence of a unique optimal stopping time in dimensions ≥ 3 for α ≠ 2.
- Optimal stopping times are characterized as hitting times to symmetric barriers.
- The problem is connected to subharmonic martingale optimal transport and duality theory.

## 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.