CONTROLLED MEASURE-VALUED MARTINGALES: A VISCOSITY SOLUTION APPROACH

• 成果类型: Article
• 署名作者: Cox, Alexander M. G.; Kallblad, Sigrid; Larsson, Martin; Svaluto-Ferro, Sara
• 署名单位: University of Bath; Royal Institute of Technology; Carnegie Mellon University; University of Verona
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP2012
• 发表日期: 2024
• 页码: 1987-2035
• 关键词: optimal transport; Bellman equation; games; EXISTENCE; SPACE; sdes

摘要

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 Ito'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.