Stochastic Perron for stochastic target games

TL;DR

This paper extends the stochastic Perron method to stochastic target games, providing a way to characterize the value function as a viscosity solution of the associated HJB equation, and establishing the dynamic programming principle.

Contribution

It introduces a novel application of the stochastic Perron method to stochastic target games, linking it to viscosity solutions and the dynamic programming principle.

Findings

Established viscosity sub- and super-solutions for the HJB equation.

Characterized the value function as a viscosity solution.

Proved the dynamic programming principle for the game.

Abstract

We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solution (super-solution) of a Hamilton-Jacobi-Bellman (HJB) equation. We then characterize the value function as a viscosity solution to the HJB equation using a comparison result and a byproduct to obtain the dynamic programming principle.