Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of non-ruin of an insurance company, in a finite-time interval is derived, assuming independent, non-identically Erlang distributed claim inter-arrival times, Permanent repository link. ., any continuous joint distribution of the claim amounts and any non-negative, non-decreasing real function, representing its premium income. In the special case when τ i ∼ Erlang (g i , λ) , i = 1, 2, . . . it is shown that our main result yields a formula for the probability of non-ruin expressed in terms of the classical Appell polynomials. We give another special case of our non-ruin probability formula for. ., i.e., when the inter-arrival times are non-identically exponentially distributed and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when τ i ∼ Erlang(1, λ), i = 1, 2, . . .. The main result is extended further to a risk model in which inter-arrival times are dependent random variables, obtained by randomizing the Erlang shape or/and rate parameters. We give also some useful auxiliary results which characterize and express explicitly (and recurrently) the exponential Appell polynomials which appear in our finite time non-ruin probability formulae.

On finite-time ruin probabilities in a generalized dual risk model with dependence

“…Another important contribution of the paper is the result of Lemma B.1, which generalizes a previous result of Ignatov and Kaishev (2012), obtained for the case of Erlang claim amounts to the case of claim sizes following a linear combination of exponentials. As noted in section 2.1, due to the connection established by Lemma 2.1, the numerical properties of formula (9) (respectively (7) and (5)) are similar to the numerical properties of the ruin probability formulas in the (direct) insurance risk model considered recently by Dimitrova et al (2013).…”

“…This result is new and can be viewed as a generalization of Theorem 2.1 of Ignatov and Kaishev (2012) obtained for the case of Erlang(g i , λ i ) inter-arrival times, i.e. when α ij ≡ 1, g i ≡ m i and λ ij ≡ λ i in the definition (4).…”

“…Here, similarly to the derivation in Section 2.1 in Ignatov and Kaishev (2012), we derive a formula for the finite-time survival probability in the insurance risk model with inter-arrival times distributed as a linear combination of exponential random variables. We suppress the use of subscript ins to simplify notations.…”

“…formulae (7), (9) and (11)) which have not been previously known in the dual risk modelling literature. Second, we prove Lemma B.1 which is new and generalizes the result of Ignatov and Kaishev (2012) for the finite-time ruin probability in the insurance risk model, to the case of inter-arrival times distributed as linear combinations of exponential random variables. This result is important on its own right, and combining it with Lemma 2.1 has allowed us to establish Proposition 2.2 which gives a new ruin probability formula for the dual risk model, also with arbitrarily distributed capital gains.…”

“…Our demonstration of the connection is different, more straightforward and applies under more general assumptions. By revisiting the formulas of survival probability in two reasonably general insurance risk models considered by Ignatov and Kaishev (2004) and Ignatov and Kaishev (2012), we obtain explicit formulas for the finite-time survival probability in our generalized dual risk model for exponential and Erlang capital gains, in terms of classical Appell polynomials and the so-called exponential Appell polynomials, respectively.…”