Stopping of functionals with discontinuity at the boundary of an open set

TL;DR

This paper investigates the properties of value functions and optimal stopping times for functionals with boundary-related discontinuities, providing a numerical approximation method and establishing the existence of near-optimal stopping times.

Contribution

It introduces a generalized penalty method for approximating value functions in the presence of boundary discontinuities and proves the existence of optimal or near-optimal stopping times.

Findings

Developed a numerical algorithm for value function approximation

Proved existence of optimal or epsilon-optimal stopping times

Extended analysis to general Feller-Markov processes

Abstract

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set $\mathcal{O}$. The stopping horizon is either random, equal to the first exit from the set $\mathcal{O}$, or fixed: finite or infinite. The payoff function is continuous with a possible jump at the boundary of $\mathcal{O}$. Using a generalization of the penalty method we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or $\epsilon$-optimal stopping times.