Optimal proportional reinsurance policies for diffusion models with transaction costs

• Published in: Insurance, mathematics & economics mathematics & economics, 1998-05, Vol.22 (1), p.41-51
• Main Authors: Højgaard, Bjarne, Taksar, Michael
• Format: Article
• Language: English
• Subjects: Capital market; Diffusion models; Economic theory; HJB equation; Insurance; Reinsurance; Statistical analysis; Stochastic control; Stochastic control theory; Stochastic models; Stochastic processes; Studies; Transaction costs
• ISSN: 0167-6687; 1873-5959
• DOI: 10.1016/S0167-6687(98)00007-9

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 R t π is governed by a SDE d R t π = ( μ − (1 − a π ( t)) λ d t + a π ( t) σ d W t , where W t is a standard Brownian motion, μ, σ > 0 are constants and λ ≥ μ. The stochastic process a π ( t) satisfying 0 X≤ a π ( t) ≤ 1 is the control process, where 1 − a π ( 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 V π ( x) = E ∫ τ π 0 e − ct R π t d t, where c > 0, τ π 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 ≥ 2 μ, the optimal policy is not to reinsure, and if μ > λ > 2 μ, the optimal fraction of reinsurance as a function of the current reserve monotonically increases from 2( λ − μ)/ λ to 1 on (0, x 1) for some constant x 1 determined by exogenous parameters.