On a double barrier hybrid dividend strategy in a compound Poisson risk model with stochastic income

Abstract

The paper features a hybrid dividend payment strategy on an insurance surplus with stochastic income. The hybrid dividend strategy works as follows: A delay-clock starts whenever the surplus up-shoots, due to a premium arrival, to a level between barriers ‘k’ and ‘$\ell$’, such that $0 \leqslant u \leqslant k \leqslant \ell < \infty$. Further, the clock restarts along with each premium or claim arrival, provided the surplus still lies between ‘k’ and ‘$\ell$’. However, when the surplus exits to a level outside the range between two barriers, the clock terminates. The insurer pays a dividend of an amount exceeding the level ‘k’ to shareholders on two scenarios. In one situation, the insurer pays dividend if delay-clock alarms before a premium or claim arrival. In the other situation, the insurer pays dividend instantaneously if surplus exceeds the second barrier ‘$\ell$’. The aggregate premiums and claims are both following mutually independent compound Poisson processes, both having exponential amount sizes which are further independent to corresponding inter-arrival times. A scheme for determining analytic expressions for piecewise Volterra integral equations, with boundary conditions, which are satisfied by the expected discounted dividend paid before ruin and the Gerber–Shiu function are discussed in detail when the delay-clock also follows an exponential distribution. We discuss a numerical example to show the tractability of the scheme and to brief the impact of hybrid dividend strategy over the classical De Finetti barrier strategy for paying dividends.