Branched harmonic majorants: representations for multidimensional optimal stopping

TL;DR

This paper introduces branched harmonic majorants to construct the least superharmonic majorant of a function in multiple dimensions, providing a new approximation scheme for optimal stopping problems related to Brownian motion.

Contribution

It develops a novel constructive approximation scheme for multidimensional optimal stopping problems using branched harmonic majorants, extending classical theories.

Findings

Identifies the optimal stopping region as the contact set with the gain function.

Provides an explicit approximation scheme via truncation of the branching depth.

Generalizes the Dynkin–Yushkevich concave-envelope theorem to multiple dimensions.

Abstract

We construct the least superharmonic majorant of a continuous function $g$ on the $d$-dimensional unit ball ($d \geq 2$) via a canonical sequential scheme. While classical theory identifies this majorant with the value function of the optimal stopping problem for Brownian motion absorbed at the domain boundary, no comparable constructive approximation scheme has been available. We introduce branched harmonic majorants, obtained by arranging classical harmonic functions on smoothly bounded domains in a finite, depth-indexed branching structure, and prove two main results. First, the optimal stopping region is identified as the contact set between the gain function $g$ and the pointwise infimum of this family; the value function is recovered as the expected gain at the first exit time from the non-contact set. This yields a multidimensional generalisation of the Dynkin--Yushkevich concave-envelope theorem in which affine functions are replaced by branched harmonic majorants. Second, truncation in the branching depth produces a decreasing sequence of envelopes that converges pointwise to this infimum, yielding an explicit approximation scheme not present in classical formulations. Analytically, the branching structure relaxes the global majorisation constraint to a local constraint imposed on a decreasing sequence of non-contact sets, yielding a representation of the Perron envelope in terms of harmonic functions on smoothly bounded domains. Probabilistically, the construction corresponds to the sequential composition of stopping times and overcomes the localisation obstruction arising from the thinness of Brownian paths in dimensions $d \geq 2$.