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We study the optimal excess-of-loss reinsurance problem when both theintensity of the claims arrival process and the claim size distribution areinfluenced by an exogenous stochastic factor. We assume that the insurer'ssurplus is governed by a marked point process with dual-predictable projectionaffected by an environmental factor and that the insurance company can borrowand invest money at a constant real-valued risk-free interest rate $r$. Ourmodel allows for stochastic risk premia, which take into account riskfluctuations. Using stochastic control theory based on theHamilton-Jacobi-Bellman equation, we analyze the optimal reinsurance strategyunder the criterion of maximizing the expected exponential utility of theterminal wealth. A verification theorem for the value function in terms ofclassical solutions of a backward partial differential equation is provided.Finally, some numerical results are discussed.

Optimal excess-of-loss reinsurance for stochastic factor risk models