Equilibrium in a Reinsurance Dynamic Risk Setting: Optimal Portfolio Selection

This paper considers the equilibrium in a reinsurance dynamic risk setting to have the optimal portfolio selection for the insurer and reinsurance in a fixed term insurance contract which consists of reinsurance price and risk retention level. The risk process is assumed to be a diffusion approximation process of the classic Cramer–Lundberg model which is perturbed by a Brownian motion with drift. We suppose that both the insurer and reinsurer have constant absolute risk aversion preferences with risk aversion coefficients and study the optimal reinsurance models from the perspective of both the insurer and the reinsurer by maximizing the expected exponential utility of terminal wealth given the information set $$\mathcal{\{F\}}_{t}$$ at time $$t$$ using Hamilton–Jacobi–Bellman equation. To obtain the suitable insurance portfolios for the insurance and reinsurer, we use the principle of dynamic programming. Moreover, the simultaneous problems are presented to our insurance portfolio. Finally, to better illustrate the derived formulas we shall study several examples in details and investigate the effect of parameters of models on our optimization problem as well as the economic meaning behind.