Optimal proportional reinsurance policies for diffusion models with transaction costs

This paper extends the results of Højgaard and Taksar (1997a) to the case of posititve transactions costs. The setting here and in Højgaard and Taksar (1997a) is the following: When applying a proportional reinsurance policy π the reserve of the insurance company Rtπ is governed by a SDE dRtπ = (μ − (1 − aπ (t))λ dt + aπ (t)σ dWt, where Wt is a standard Brownian motion, μ, σ > 0 are constants and λ ≥ μ. The stochastic process aπ (t) satisfying 0 X≤ aπ (t) ≤ 1 is the control process, where 1 − aπ (t) denotes the fraction of all incoming claims, that is reinsured at time t. The aim of this paper is to find a policy that maximizes the return function Vπ (x) =E∫τπ0 e−ct Rπt dt, where c > 0, τπ is the time of ruin and x refers to the initial reserve. In Højgaard and Taksar (1997a) a closed form solution is found in case of λ = μ by means of Stochastic Control Theory. In this paper we generalize this method to the more general case where we find that if λ ≥ 2μ, the optimal policy is not to reinsure, and if μ > λ > 2μ, the optimal fraction of reinsurance as a function of the current reserve monotonically increases from 2(λ − μ)/λ to 1 on (0, x1) for some constant x1 determined by exogenous parameters.