2018/05/07 by Cong Wu, Jianfeng Zhang, Wu, Cong +1 · 2 citations
Economics, Econometrics and Finance · Social Sciences · #Analysis of PDEs (math.AP) #Economic theories and models #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1805.02639
openalex publication_date 2018/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 \citeDupire's functional Itô formula. This Itô formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions \citeLions4 in the state dependent case. We provided an elementary proof for this well known result in the short note \citeWZ, and the same arguments work in the path dependent setting here.