Annals of Applied Probability

Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls

Cong Wu and Jianfeng Zhang

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Abstract

The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to a certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean–Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire’s (2009) functional Itô formula. This Itô formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions in the state dependent case. We provided an elementary proof for this well known result in the short note (2017), and the same arguments work in the path dependent setting here.

Article information

Source
Ann. Appl. Probab., Volume 30, Number 2 (2020), 936-986.

Dates
Received: September 2018
Revised: April 2019
First available in Project Euclid: 8 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1591603227

Digital Object Identifier
doi:10.1214/19-AAP1521

Mathematical Reviews number (MathSciNet)
MR4108127

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 49L25: Viscosity solutions 60H30: Applications of stochastic analysis (to PDE, etc.) 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 49L20: Dynamic programming method 93E20: Optimal stochastic control

Keywords
Master equation McKean–Vlasov SDEs viscosity solution functional Itô formula path dependent PDEs Wasserstein spaces dynamic programming principle

Citation

Wu, Cong; Zhang, Jianfeng. Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls. Ann. Appl. Probab. 30 (2020), no. 2, 936--986. doi:10.1214/19-AAP1521. https://projecteuclid.org/euclid.aoap/1591603227


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