Viscosity Solutions and Mean Field Equilibria for Nonlocal Stochastic Control Under Catastrophe and Replacement-Cost Risk

Abstract

We develop a stochastic-control and mean field game framework for catastrophe insurance under stochastic replacement-cost risk. Insurer surplus follows a controlled jump diffusion in which catastrophe losses are scaled by an exogenous mean-reverting replacement-cost factor and attenuated through physical hedging. We establish well-posedness and stability of the controlled state process, derive a stopping-time dynamic programming principle, and characterize the value function as the unique viscosity solution of the associated nonlocal Hamilton–Jacobi–Bellman (HJB) equation. We then formulate strategic interaction among insurers through a coupled nonlocal HJB–Kolmogorov system and establish existence and uniqueness of mean field equilibrium under regularity and monotonicity conditions. The analysis quantifies how elevated replacement costs amplify catastrophe-loss exposure while physical hedging reduces it. Numerical results indicate stronger optimal hedging under high replacement-cost states and weak capitalization and quantify the equilibrium effects of industry-wide vulnerability.