Robust Optimal Per-loss Reinsurance Strategy for an Ambiguity-Averse Insurer

We investigate the equilibrium reinsurance strategy in an infinite reinsurance space for an ambiguity-averse insurer (AAI) under a continuous-time framework. We assume that the surplus process of the AAI follows the Cram\'er-Lundberg (C-L) model perturbed by standard Brownian motion, and the insurer invests his surplus in a risk-free asset. We present the equilibrium reinsurance strategy and its corresponding value function by solving extended Hamilton-Jacobi-Bellman (HJB) system equations, and we find that the AAI's equilibrium reinsurance strategy to maximize the time-inconsistent penalty-dependent mean-variance reward function is a combination of quota-share with excess of loss reinsurance or its dual form. Detailed numerical analyses are presented to illustrate the various effects of insurer aversion to various uncertainties and other parameters on the equilibrium reinsurance strategy and its corresponding value function.