On a class of optimal stopping problems for diffusions with discontinuous coefficients

TL;DR

This paper develops a modified PDE approach for optimal stopping problems of diffusions with discontinuous coefficients, enabling analysis where traditional regularity assumptions fail, and provides explicit solutions in special cases.

Contribution

Introduces a new modification of the free boundary problem that applies to diffusion optimal stopping problems with irregular coefficients and gain functions.

Findings

Established a general verification theorem for the modified free boundary problem.

Provided explicit solutions for the stopping problem in the no drift and no discount case.

Extended PDE methods to cases with discontinuous coefficients and irregular data.

Abstract

In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form $E_x\int_0^{\tau}e^{-\lambda s}f(X_s) ds$, $\lambda\ge0$ for one-dimensional diffusions $X$. We prove a general verification theorem which justifies the modified version of the free boundary problem. In the case of no drift and discount, the free boundary problem allows to give a complete and explicit discussion of the stopping problem.