Controlled Measure-Valued Martingales: a Viscosity Solution Approach

TL;DR

This paper studies stochastic control problems involving measure-valued martingales, establishing a viscosity solution framework for the associated Hamilton-Jacobi-Bellman equations, with applications in finance and game theory.

Contribution

It introduces a viscosity solution approach for control problems with measure-valued martingales, including an Itô's lemma for these processes and applications in various fields.

Findings

Proves the value function is the unique viscosity solution to the HJB equation.

Develops an Itô's lemma for controlled measure-valued martingales.

Connects the theory to applications in finance and game theory.

Abstract

We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We establish the `classical' results of stochastic control for these problems: specifically, we prove that the value function for the problem can be characterised as the unique solution to the Hamilton-Jacobi-Bellman equation in the sense of viscosity solutions. In order to prove this result, we exploit structural properties of the MVM processes. Our results also include an appropriate version of It\^o's lemma for controlled MVMs. We also show how problems of this type arise in a number of applications, including model-independent derivatives pricing, the optimal Skorokhod embedding problem, and two player games with asymmetric information.