A dynamic pricing game for general insurance market

Insurance contracts pricing, that is determining the risk loading added to the expected loss, plays a fundamental role in insurance business. It covers the loss from adverse claim experience and generates a profit. As market competition is a key component in the pricing exercise, this paper proposes a novel dynamic pricing game model with multiple insurers who are competing with each other to sell insurance contracts by controlling their insurance premium. Different with the existing works assuming deterministic surplus/loss, we consider stochastic surplus and adopt the linear Brownian motion model, i.e., a diffusion approximation to the classical Cramér–Lundberg model, for the aggregate claim amount. The risk exposure of an insurer is assumed to be affected by all insurers in the market. By solving a system of Hamilton–Jacobi–Bellman (HJB) equations, Nash equilibrium premium strategies are explicitly obtained for the insurers who are aiming to maximize their expected terminal exponential utilities. The representation form of the equilibrium strategies relates to the so-called M-matrix, which appears in many economic models. To investigate the robustness of equilibrium pricing strategies under model uncertainty, we further extend the model by allowing insurers to perceive ambiguity towards the aggregate claim loss. Closed-form expression for the robust premium strategies are obtained and comparative statics are carried out for model parameters.