🤖 AI Summary
This study addresses dynamic physical hedging under the joint risks of catastrophe losses and stochastic reconstruction costs by formulating a controlled jump-diffusion mean-field game model. We establish the viscosity solution and relaxed equilibrium theory for the associated non-local HJB-Kolmogorov coupled system, proving the existence and uniqueness of the equilibrium. The analysis elucidates the intrinsic mechanisms through which reconstruction cost uncertainty and population vulnerability influence optimal strategies. Furthermore, numerical simulations demonstrate that capitalization levels significantly alter hedging behaviors and confirm the tail robustness of the proposed strategies. Collectively, this work provides a novel theoretical framework for insurance hedging within complex risk environments characterized by systemic interactions and discontinuous dynamics.
📝 Abstract
We study dynamic physical hedging for insurers exposed jointly to catastrophe losses and stochastic reconstruction costs. Surplus evolves as a controlled jump diffusion whose loss amplitude combines marked catastrophe severity, an exogenous mean-reverting cost factor, and endogenous mitigation. We establish well-posedness, moment and stability estimates, and a stopping-time dynamic programming principle, and prove that the value function is the unique viscosity solution of the resulting nonlocal Hamilton-Jacobi-Bellman (HJB) equation Strategic interaction is introduced through a mean field game (MFG) with reduced-form vulnerability costs, yielding a coupled backward-forward HJB-Kolmogorov system. We establish relaxed equilibrium existence, Markovian realization, and uniqueness under appropriate compactness and monotonicity conditions. Numerical experiments show that reconstruction costs and capitalization materially affect optimal hedging and that cross-sectional vulnerability alters equilibrium costs. Tail-family robustness calculations further assess the sensitivity of these conclusions to alternative catastrophe-severity specifications.