Minimum curvature flow and martingale exit times

TL;DR

This paper characterizes the maximum time a normalized martingale can stay within a convex body using a geometric flow called minimum curvature flow, linking stochastic processes with geometric evolution equations.

Contribution

It introduces the concept of minimum curvature flow to determine the deterministic exit time for martingales inside convex bodies, connecting stochastic and geometric flow theories.

Findings

The maximum deterministic exit time equals the time for the boundary to reach the initial point under minimum curvature flow.

Establishes a viscosity solution framework for the problem.

Links the flow to Ambrosio–Soner mean curvature flow of the convex body's skeleton.

Abstract

We study the following question: What is the largest deterministic amount of time $T_*$ that a suitably normalized martingale $X$ can be kept inside a convex body $K$ in $\mathbb{R}^d$? We show, in a viscosity framework, that $T_*$ equals the time it takes for the relative boundary of $K$ to reach $X(0)$ as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio--Soner codimension-$(d-1)$ mean curvature flow of the $1$-skeleton of $K$. Our results are obtained by a mix of probabilistic and analytic methods.