An alternating risk reserve process. II. (English) Zbl 1198.91085

Boxma, O. J.; Jönsson, H.; Resing, J. A. C.; Shneer, S.

Markov Process. Relat. Fields 16, No. 2, 425-446 (2010).

Summary: We consider an alternating risk reserve process with a threshold dividend strategy. The process can be in two different states and the state of the process can only change at the arrival instants of an independent Poisson observer. Whether or not a change then occurs depends on the value of the risk reserve w.r.t. the barrier. If at such an instant the capital is below the threshold, the system is set to state 1 (paying no dividend), and if the capital is above the threshold, the system is set to state 2 (paying dividend). In each of the two states, the process is described by different premium rates, Poisson claim arrival intensities, and claim size distributions. For this model we determine the survival probabilities, distinguishing between the initial state being 1 or 2, and the process starting below or above the barrier. In the case of exponentially distributed claim sizes, survival probabilities are found by solving a system of integro-differential equations. In the case of generally distributed claim sizes, they are expressed in the survival probabilities of the corresponding standard risk reserve processes. We perform several numerical experiments, including a comparison with the case in which state changes can only occur just after claim arrival instants; that case is treated in Part I.

MSC:

91B30

Risk theory, insurance (MSC2010)

Keywords:

insurance risk models; alternating risk reserve process; survival probabilities