Optimal Brownian Stopping When the Source and Target Are Radially Symmetric Distributions

DOI10.1137/19M1270513zbMATH Open1450.60031arXiv1906.11635MaRDI QIDQ5130894

Young-Heon Kim, Tongseok Lim, Nassif Ghoussoub

Publication date: 29 October 2020

Published in: SIAM Journal on Control and Optimization (Search for Journal in Brave)

Abstract: Given two probability measures $mu,u$ on $mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $au$ that maximize/minimize the cost functional $mathbbE|B_0-B_{a}u{|}^{alpha}$, $alpha>0$, where $(B_t{)}_t$ is Brownian motion with initial law $mu$ and with final distribution --once stopped at $au$-- equal to $u$. Under the assumption of radial symmetry on $mu$ and $u$, we show that in dimension $dgeq3$ and $alphaeq2$, 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.

Full work available at URL: https://arxiv.org/abs/1906.11635

zbMATH Keywords

monotonicityradial symmetrySkorokhod embeddingsubharmonic martingale optimal transport

Mathematics Subject Classification ID

Probability distributions: general theory (60E05) Brownian motion (60J65) Stopping times; optimal stopping problems; gambling theory (60G40) Optimal transportation (49Q22)

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