Integral representation of subharmonic functions and optimal stopping with random discounting

TL;DR

This paper develops an integral representation for positive subharmonic functions of one-dimensional diffusions, enabling explicit solutions to optimal stopping problems with random discounting by transforming the underlying process.

Contribution

It introduces a novel integral equation representation for subharmonic functions and applies it to solve optimal stopping problems with random discounting explicitly.

Findings

Integral representation for positive subharmonic functions established.

Constructed measure transformations that alter diffusion behavior.

Provided explicit solutions to optimal stopping problems with random discounting.

Abstract

An integral representation result for strictly positive subharmonic functions of a one-dimensional regular diffusion is established. More precisely, any such function can be written as a linear combination of an increasing and a decreasing subharmonic function that solve an integral equation \[ g(x)=a + \int v(x,y)\mu_A(dy) + \kappa s(x), \] where $a>0$, $\kappa \in \mathbb{R}$, $s$ is a scale function of the diffusion, $\mu_A$ is a Radon measure, and $v$ is a kernel that is explicitly determined by the scale function. This integral equation in turn allows one construct a pair $(g,A)$ such that $g$ is a subharmonic function, $A$ is a continuous additive functional with Revuz measure $\mu_A$ and $g(X)\exp(-A)$ is a local martingale. The changes of measures associated with such pairs are studied and shown to modify the long term behaviour of the original diffusion process to exhibit transience. Theory is illustrated via examples that in particular contain a sequence of measure transformations that render the diffusion irregular in the limit by breaking the state space into distinct regions with soft and hard borders. Finally, the theory is applied to find an "explicit" solution to an optimal stopping problem with random discounting.