A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls

TL;DR

This paper establishes a weak dynamic programming principle for zero-sum stochastic differential games with unbounded controls, linking the game’s value to a viscosity solution of a nonlinear PDE.

Contribution

It introduces a weak dynamic programming principle for such games, extending the theory to unbounded control settings and connecting it with viscosity solutions.

Findings

Players' value functions satisfy a weak dynamic programming principle.

Value functions are viscosity solutions of the associated PDE.

The framework handles unbounded controls in stochastic differential games.

Abstract

We analyze a zero-sum stochastic differential game between two competing players who can choose unbounded controls. The payoffs of the game are defined through backward stochastic differential equations. We prove that each player's priority value satisfies a weak dynamic programming principle and thus solves the associated fully non-linear partial differential equation in the viscosity sense.