Asymptotics for the Finite Time Ruin Probability in the Renewal Model with Consistent Variation

Abstract

This paper investigates the finite time ruin probability in the renewal risk model. Under some mild assumptions on the tail probabilities of the claim size and of the inter-occurrence time, a simple asymptotic relation is established as the initial surplus increases. In particular, this asymptotic relation is requested to hold uniformly for the horizon varying in a relevant infinite interval. The uniformity allows us to consider that the horizon flexibly varies as a function of the initial surplus, or to change the horizon into any nonnegative random variable as long as it is independent of the risk system.

Tail behavior of random sums under consistent variation with applications to the compound renewal risk model

“…For recent contributions in case of the classical renewal risk model, i.e. when N 1 = N 2 = · · · = 1, see Tang (2004), Leipus and Šiaulys (2007). For example, in the case of classical renewal risk model Tang (2004, Theorem 3.1) obtained the following result:…”

“…F and positive y, set According to the terminology of Bingham et al (1987), the quantities J − F and J + F are the lower and upper Matuszewska indices of the function (F(x)) −1 , x ≥ 0. As in Tang (2004), we call J − F and J + F respectively the lower and upper Matuszewska indices of d.f. F(x).…”

Large-deviation probabilities for maxima of sums of subexponential random variables with application to finite-time ruin probabilities

“…By (18), (19), and the arbitrariness of δ > 0, we obtain the uniform asymptotic relation (11). This completes the proof of Theorem 3.1.…”

“…Hence, his result works essentially only for the case of Pareto-like claim sizes. Leipus andŠiaulys [15] extended the result of Tang [11] to a larger class by adding some strong restrictions on the hazard rate function of B with the uniformity the same as in our theorem. Clearly, for the compound Poisson model, our theorem extends the result of Tang [11] to the whole class S * .…”

“…Tang [11] established (11) in the renewal model under the assumption, among others, that the distribution B is consistently varying tailed. Hence, his result works essentially only for the case of Pareto-like claim sizes.…”