Extremes on the discounted aggregate claims in a time dependent risk model

Abstract

This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables. Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed random variables. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims.A simulation study is performed in order to validate the results obtained in the free interest risk model.

Asymptotics for risk capital allocations based on Conditional Tail Expectation

“…For the dependence structures discussed by Asimit and Badescu (2010) and Li et al (2010) with strongly equivalent tails, both Assumptions 3.1 and 3.2 are satisfied. It is not difficult to prove that a bivariate random vector with strongly equivalent tails and dependence structure given by the Marshall-Olkin copula (see, e.g., Nelsen, 1999, page 46) satisfies only Assumption 3.1.…”

“…It is known that, for both Fréchet and Gumbel cases, the CTE and VaR of a single risk are proportional for a high confidence level (see Asimit and Badescu, 2010 The remainder of the paper is organized as follows. The main results under asymptotic dependence and asymptotic independence are formulated and proved in Sections 2 and 3, respectively.…”

Ruin probabilities for a regenerative Poisson gap generated risk process

“…After that comes a long part ω (2) with mean m + . For the final part ω (3) , two possibilities occur: The next interarrival time is long and the cycle terminates. This gives an additional contribution to the mean of p + m + .…”

Tail asymptotic expansions for L-statistics

“…Typically, the properties of S n (c) are derived when n becomes large, i.e., n → ∞, see Beirlant and Teugels [8], Ladoucette and Teugels [19], Ladoucette and Teugels [20], Ladoucette and Teugels [21] which also present several financial and insurance applications. In other applications, for instance when modelling the financial losses of n portfolios, it is not possible to change the number of portfolios under investigation, and therefore of interest is the tail asymptotic behaviour of S n (c) for each fixed n. The recent contribution Asimit et al [5] (see also Asimit and Badescu [2], Asimit and Jones [3], Asimit and Jones [4]) shows that under weak asymptotic conditions P (S n (c) > x) ∼ P (c 1 X n,n > x) as x → ∞, which means that the maximum controls the asymptotic behaviour of the L-statistics S n (c). For applications, it is of interest to know the speed of convergence to 0 of ∆(x) = P (S n (c) > x) − P (c 1 X n,n > x), i.e., how well the maximum risk controls the L-statistics S n (c).…”