Optimal proportional reinsurance policies for diffusion models with transaction costs

This paper extends the results of Hojgaard and Taksar (1997a) to the case of posititve transactions costs. The setting here and in Hojgaard and Taksar (1997a) is the following: When applying a proportional reinsurance policy ir the reserve of the insurance company (R-t(pi)) is governed by a SDE dR(t)(pi) = (mu -(1 -a(pi)(t))lambda dt + a(pi) (t)sigma dW(t), where {W-t} is a standard Brownian motion, mu, sigma > O are constants and lambda greater than or equal to mu. The stochastic process {a(pi) (t)} satisfying O less than or equal to a(pi) (t) less than or equal to 1 is the control process, where 1 -a(pi)(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-pi(x) = E integral(o)(tau pi) e(-ct)R(t)(pi) dt, where c > O, tau(pi) is the time of ruin and x refers to the initial reserve. In Hojgaard and Taksar (1997a) a closed form solution is found in case of lambda = CL by means of Stochastic Control Theory. In this paper we generalize this method to the more general case where we find that if lambda > 2 mu, the optimal policy is not to reinsure, and if mu < lambda < 2 mu, the optimal fraction of reinsurance as a function of the current reserve monotonically increases from 2(lambda-mu)/lambda to 1 on (O, x(1)) for some constant x(1) determined by exogenous parameters. (C) 1998 Elsevier Science B.V. All rights reserved.