Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications

TL;DR

This paper establishes comparison results for quadratic growth solutions of second-order Bellman-Isaacs equations and applies these results to characterize the value function in stochastic control problems.

Contribution

It provides new comparison theorems for quadratic growth viscosity solutions and characterizes the value function as the unique solution of the associated dynamic programming equation.

Findings

Comparison result for quadratic growth solutions

Characterization of the value function as unique viscosity solution

Applicability to stochastic control problems with unbounded controls

Abstract

In this paper, we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.