SIAM Journal on Control and Optimization, Volume 58, Issue 5, Page 2765-2789, January 2020.
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 nonrandomized 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.

中文翻译：

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源和目标是径向对称分布时的最优布朗停止

SIAM控制与优化杂志，第58卷，第5期，第2765-2789页，2020年1月。
给定两个概率度量$ \ mu，\ nu $ on $ \ mathbb {R} ^ d $，按次谐波顺序，我们描述了最优停止时间$ \ tau $，最大化/最小化成本函数$ \ mathbb {E} | B_0-B_ \ tau | ^ {\ alpha} $，$ \ alpha> 0 $，其中$（B_t）_t $是布朗运动最初的法律为\\ mu $，而最终的分配为一次，即停在$ \ tau $-等于$ \ nu $。在$ \ mu $和$ \ nu $呈径向对称的假设下，我们表明在维$ d \ geq 3 $和$ \ alpha \ neq 2 $上，存在一个由非随机停止时间给出的唯一最优解作为到达适当对称障碍的击球时间。我们还将这个问题与次谐波mar的最优运输问题联系起来，并建立了对偶结果。