A Probabilistic Formula for the Concave Hull of a Function

Abstract

Let $D$ be a compact, convex domain in $d$-dimensional Euclidean space and let $f$ be a nonnegative real-valued function defined on $D$. The classical optimal stopping problem is to find a stopping time $\tau^\ast$ that attains the supremum $\nu(x) = \sup_\tau EE_xf(B(\tau))$. Here, $B$ is a $d$-dimensional Brownian motion with absorption on the boundary of $D$ and the supremum is over all stopping times. It is well known that $\nu$ is characterized as the smallest superharmonic majorant of $f$. In this paper, we modify this problem by allowing $B$ to be essentially any drift-free diffusion (with absorption, as before, on the boundary of $D$). For example, it could be a Brownian motion diffusing on some lower dimensional affine set. In addition, one is allowed to switch among these diffusions at any time. The problem is to find a stopping time and a switching strategy that together attain the supremum over all stopping times and all switching strategies. For this problem, we show that $\nu$ is characterized as the smallest concave majorant of $f$. The domain $D$ can be decomposed into a disjoint union of relatively open convex sets on each of which the function $\nu$ is affine. Furthermore, the union of the zero-dimensional convex sets is contained in the set on which $\nu = f$. An optimal switching strategy is any strategy that at all times diffuses in the affine hull of the current convex set. When the diffusion reaches the boundary of the current convex set, it will lie on a lower dimensional convex set and must then diffuse on the affine hull of this new set. This process continues until the set on which $\nu = f$ is reached, which is the optimal stopping time.