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Optimal stopping for multidimensional Brownian motion

John Moriarty

TL;DR

This paper analyzes optimal stopping for a multidimensional Brownian motion in the unit ball, with gain $g$ of compact support and boundary absorption. It develops a potential-theoretic framework that represents the value function $V$ as the pointwise infimum of dominating Dirichlet Laplacian potentials, first under a classical regime ($M= abla$) and then in a branching setting to handle more complex continuation structures. The authors introduce a pathwise extension algorithm and a hierarchy of branching potentials, culminating in two main theorems: a classical representation $V=w_1$ under a regularity condition and a generalized representation $V=w^*$ via branching potentials with a Schauder-estimate-based argument. The work bridges stochastic optimal stopping, potential theory, and differential-geometry approximations to address multidimensional free-boundary problems and offers a constructive, martingale-based method for approximating the value function and optimal policy in higher dimensions.

Abstract

We address the optimal stopping of multidimensional Brownian motion in a bounded domain. Under a geometric condition, we provide a simple proof that the value function is the pointwise infimum of Dirichlet Laplacian potentials dominating the gain function. Weakening this condition, approximations from differential geometry and potential theory provide a similar result when the infimum is taken over a branching set of piecewise potentials.

Optimal stopping for multidimensional Brownian motion

TL;DR

This paper analyzes optimal stopping for a multidimensional Brownian motion in the unit ball, with gain of compact support and boundary absorption. It develops a potential-theoretic framework that represents the value function as the pointwise infimum of dominating Dirichlet Laplacian potentials, first under a classical regime () and then in a branching setting to handle more complex continuation structures. The authors introduce a pathwise extension algorithm and a hierarchy of branching potentials, culminating in two main theorems: a classical representation under a regularity condition and a generalized representation via branching potentials with a Schauder-estimate-based argument. The work bridges stochastic optimal stopping, potential theory, and differential-geometry approximations to address multidimensional free-boundary problems and offers a constructive, martingale-based method for approximating the value function and optimal policy in higher dimensions.

Abstract

We address the optimal stopping of multidimensional Brownian motion in a bounded domain. Under a geometric condition, we provide a simple proof that the value function is the pointwise infimum of Dirichlet Laplacian potentials dominating the gain function. Weakening this condition, approximations from differential geometry and potential theory provide a similar result when the infimum is taken over a branching set of piecewise potentials.
Paper Structure (9 sections, 15 theorems, 44 equations, 2 figures, 1 algorithm)

This paper contains 9 sections, 15 theorems, 44 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Probabilistic setting
  4. Potentials
  5. Classical potentials
  6. Branching potentials
  7. Pathwise extensions
  8. Preliminary results
  9. Main result

Key Result

Lemma 1

For $M \geq \frac{g^*}{\delta}$, where $\delta := d_H(\mathop{\mathrm{\mathrm{supp}}}\nolimits(g), \partial \Lambda) > 0$, the function $w_1$ of Definition def:pb4a satisfies $w_1 = 0$ on $\partial{\Lambda}$.

Figures (2)

  • Figure 1: Visualisation of a piecewise potential $h \in \mathcal{H}_3$ satisfying $h \geq g$, for $d=2$. Each piece $h_i$ is a potential lying in $\mathcal{H}_0$, and they are stitched together recursively. The upper panel shows domains and sample paths, and the lower panel plots cross sections of the pieces $h_i$, which must each dominate the gain function $g$. See Remark \ref{['rem:cap']} for details.
  • Figure 2: Cross section of the approximation used in the proof of Theorem \ref{['pro:representation_4param']}, with $d=2$ (cf. Figure \ref{['fig:1']}). The envelope $w^*$ of $g$ is indicated by square markers, and the free boundary between the two is indicated by triangles. The envelope $w(\cdot,M)$ of $g$ (circular markers) approximates $w^*$ from above. The cross section of the smooth approximation $p$ to the free boundary is indicated by stars. The gain function $G$ (faint dotted line) is sandwiched between $g$ and $w(\cdot,M)$.

Theorems & Definitions (34)

  • Definition 1: Domains $\Gamma$
  • Definition 2: Potentials $\mathcal{H}_0$, $\mathcal{H}_1$
  • Definition 3: Lower envelopes
  • Lemma 1: Boundary behaviour
  • proof
  • Remark 1
  • Theorem 1
  • Lemma 2: Approximate excessivity
  • proof
  • Lemma 3
  • ...and 24 more