Stochastic Perron's method and verification without smoothness using viscosity comparison: obstacle problems and Dynkin games

TL;DR

This paper extends Stochastic Perron's method to double obstacle problems in Dynkin games, establishing the existence of a game value and optimal strategies without requiring smoothness, using viscosity solutions and comparison principles.

Contribution

It introduces a novel application of Stochastic Perron's method to non-linear double obstacle problems in Dynkin games, demonstrating existence and characterization of the value without smoothness assumptions.

Findings

Constructed viscosity sub- and super-solutions bounding the game value

Proved the game has a unique continuous viscosity solution under comparison

Identified optimal strategies as first hitting times of stopping regions

Abstract

We adapt the Stochastic Perron's method in Bayraktar and Sirbu (ArXiv: 1103.0538) to the case of double obstacle problems associated to Dynkin games. We construct, symmetrically, a viscosity sub-solution which dominates the upper value of the game and a viscosity super-solution lying below the lower value of the game. If the double obstacle problem satisfies the viscosity comparison property, then the game has a value which is equal to the unique and continuous viscosity solution. In addition, the optimal strategies of the two players are equal to the first hitting times of the two stopping regions, as expected. The (single) obstacle problem associated to optimal stopping can be viewed as a very particular case. This is the first instance of a non-linear problem where the Stochastic Perron's method can be applied successfully.