Optimal variance stopping with linear diffusions

TL;DR

This paper investigates the problem of maximizing the variance of a linear diffusion through an innovative approach that models it as a convex two-player zero-sum game, revealing new solution structures.

Contribution

It introduces a novel game-theoretic framework for variance optimization in linear diffusions and characterizes optimal stopping times as mixtures of hitting times.

Findings

Optimal stopping times are mixtures of two hitting times.

The problem can be formulated as a convex two-player zero-sum game.

The approach offers insights for more complex non-linear problems.

Abstract

We study the optimal stopping problem of maximizing the variance of an unkilled linear diffusion. Especially, we demonstrate how the problem can be solved as a convex two-player zero-sum game, and reveal quite surprising application of game theory by doing so. Our main result shows that an optimal solution can, in general case, be found among stopping times that are mixtures of two hitting times. This and other revealed phenomena together with suggested solution methods could be helpful when facing more complex non-linear optimal stopping problems. The results are illustrated by a few examples.