April 2024 Controlled measure-valued martingales: A viscosity solution approach
Alexander M. G. Cox, Sigrid Källblad, Martin Larsson, Sara Svaluto-Ferro
Author Affiliations +
Ann. Appl. Probab. 34(2): 1987-2035 (April 2024). DOI: 10.1214/23-AAP2012

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 Itô’s formula 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.

Funding Statement

The second author gratefully acknowledges financial support from the Swedish Research Council (VR) under grant 2020-03449.
The fourth author gratefully acknowledges financial support by the Vienna Science and Technology Fund (WWTF) under grant MA16-021.

Citation

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Alexander M. G. Cox. Sigrid Källblad. Martin Larsson. Sara Svaluto-Ferro. "Controlled measure-valued martingales: A viscosity solution approach." Ann. Appl. Probab. 34 (2) 1987 - 2035, April 2024. https://doi.org/10.1214/23-AAP2012

Information

Received: 1 August 2021; Revised: 1 March 2023; Published: April 2024
First available in Project Euclid: 3 April 2024

MathSciNet: MR4728162
Digital Object Identifier: 10.1214/23-AAP2012

Subjects:
Primary: 49L25 , 58J65 , 60G57 , 93E20
Secondary: 46T05‎ , 58C20 , 60G48 , 91A27 , 91G20

Keywords: Itô’s formula , measure-valued martingales , stochastic optimal control , viscosity solutions

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.34 • No. 2 • April 2024
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