Robust optimal reinsurance and investment with inflation risk: A game-theoretic approach and explicit solutions

This paper investigates robust optimal reinsurance and investment for an insurer facing both inflation risk and model ambiguity. The surplus is described by a diffusion approximation of the Cramér–Lundberg model, and purchasing-power risk is incorporated through a mean-reverting inflation factor. Specifically, the log-inflation index is modeled by an Ornstein–Uhlenbeck process, which yields a two-dimensional real-wealth system and captures state-dependent inflation effects absent under geometric Brownian inflation. Model ambiguity is represented by an adversarial probability distortion with an entropy penalty, allowing the adversary to distort the insurance, financial, and inflation channels in a unified framework. The decision problem is formulated as a zero-sum stochastic differential game. We establish a rigorous duality between the entropy-penalized robust formulation and a risk-sensitive control problem, and derive the associated Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation. Our main theoretical contribution is a tractable characterization of the robust optimal strategies in a finite-horizon setting with non-zero interest rates. By an exponential-quadratic transformation, the nonlinear HJBI equation is reduced to a system of ordinary differential equations of Riccati type, which yields feedback-form optimal reinsurance and investment rules featuring an explicit inflation-hedging component and a novel mean-reversion hedging demand. We further show that the solution collapses to the geometric-Brownian-inflation benchmark as the mean-reversion speed tends to zero. Numerical experiments illustrate that ambiguity aversion amplifies effective risk aversion, strengthens reinsurance demand, and induces a pronounced flight-to-safety effect, with substantially different hedging behavior under mean-reverting versus non-mean-reverting inflation.