OPTIMAL PROPORTIONAL REINSURANCE POLICIES FOR DIFFUSION-MODELS WITH TRANSACTION COSTS

This paper extends the results of Hojgaard and Taksar (1997a) to the c ase of posititve transactions costs. The setting here and in Hojgaard and Taksar (1997a) is the following: When applying a proportional rein surance policy ir the reserve of the insurance company (R-t(pi)) is go verned by a SDE dR(t)(pi) = (mu -(1 -a(pi)(t))lambda dt + a(pi) (t)sig ma 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 proc ess {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 f raction of all incoming claims, that is reinsured at time t. The aim o f 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 a nd 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 reser ve monotonically increases from 2(lambda-mu)/lambda to 1 on (O, x(1)) for some constant x(1) determined by exogenous parameters. (C) 1998 El sevier Science B.V. All rights reserved.