Optimal Brownian Stopping between radially symmetric marginals in general dimensions

Given an initial (resp., terminal) probability measure $\mu$ (resp., $\nu$) on ${\mathbb{R}}^d$, we characterize those optimal stopping times $\tau$ that maximize or minimize the functional $\mathbb{E}\mathrm{\mid }{B}_0-B_{\tau }{\mathrm{\mid }}^{\alpha }$, $\alpha >0$, where $(B_t{)}_t$ is Brownian motion with initial law $B_0\sim \mu$ and with final distribution --once stopped at $\tau$– equal to $B_\tau\sim \nu$. The existence of such stopping times is guaranteed by Skorohod-type embeddings of probability measures in “subharmoic order” into Brownian motion. This problem is equivalent to an optimal mass transport problem with certain constraints, namely the optimal subharmonic martingale transport. Under the assumption of radial symmetry on $\mu$ and $\nu$, we show that the optimal stopping time is a hitting time of a suitable barrier, hence is non-randomized and is unique.