The Asymptotic Estimate of Ruin Probability Under a Class of Risk Model in the Presence of Heavy Tails

In contrast with the classical Cramér–Lundberg model where the premium process is a linear function of time, we consider the ruin probability under the risk model where the aggregate premium consists of both a compound Poisson process and a linear process of time. Moreover, a constant interest force is also taken into account in our model. We restrict ourselves to the case where the claim size is heavy-tailed, i.e., the equilibrium distribution function of the claim size belongs to a wide subclass of the subexponential distribution. An asymptotic formula for the ruin probability is obtained by using the similar method of Kalashnikov and Konstantinides (2000 Kalashnikov , V. , Konstantinides , D. ( 2000 ). Ruin under interest force and subexponential claims: a simple treament . Insur. Math. Econ. 27 : 145 – 149 .Crossref], Web of Science ®] , Google Scholar]). The asymptotic formula we get here is the same as the one in Asmussen (1998 Asmussen , S. ( 1998 ). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distribution and first passage probabilities . Ann. Appl. Probab. 8 : 354 – 374 .Crossref], Web of Science ®] , Google Scholar]), Klüppelberg and Stadtmüller (1998 Klüppelberg , C. , Stadtmüller , U. ( 1998 ). Ruin probabilities in the presence of heavy-tails and interest rates . Scand. Actuarial J. 1 : 49 – 58 .Taylor & Francis Online] , Google Scholar]), and Kalashnikov and Konstantinides (2000 Kalashnikov , V. , Konstantinides , D. ( 2000 ). Ruin under interest force and subexponential claims: a simple treament . Insur. Math. Econ. 27 : 145 – 149 .Crossref], Web of Science ®] , Google Scholar]) which did not consider the stochastic premium.