On the discounted penalty function in a Markov-dependent risk model

Abstract

We present a unified approach to the analysis of several popular models in collective risk theory. Based on the analysis of the discounted penalty function in a semi-Markovian risk model by means of Laplace-Stieltjes transforms, we rederive and extend some recent results in the field. In particular, the classical compound Poisson model, Sparre Andersen models with phase-type interclaim times and models with causal dependence of a certain Markovian type between claim sizes and interclaim times are contained as special cases.

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

“…Some research has also been performed beyond the Sparre Andersen assumption of independence of the times between consecutive claim arrivals. Thus, risk models in which an appropriate dependence structure is imposed on the claim inter-arrival times and claim sizes, has been considered in [1], assuming the premium income function, h(t) = u + ct, and also in [4].…”

“…are the consecutive inter-arrival times of the claims. We will also assume that the sequence W 1 We further assume that the claim inter-arrival times τ i , i = 1, 2, . .…”

On Simple Ruin Expressions in Dependent Sparre Andersen Risk Models

“…According to our construction, J stays in state 1 for an Exp(µ 1 ) time, and moves to the states 1 or 2 with probabilities p and 1 − p respectively, where p = P (A 1 < a). Similarly, it stays in state 2 for an Exp(µ 2 ) time, and moves to the states 1 or 2 with probabilities p and 1 − p. The moves of J into state 1 (irrespective of the previous state) cause a jump of X(t) distributed as cA 1 given A 1 < a, and the moves into state 2 cause a jump of X(t) distributed as cA 1 given A 1 ≥ a (this is the analogous interpretation to the one in [4] for a risk model with dependence between claims and subsequent inter-occurrence times). Let A (1) and A (2) denote random variables distributed as cA 1 given A 1 < a and A 1 ≥ a, respectively.…”

Dividend Payments and Related Problems in a Markov-Dependent Insurance Risk Model under Absolute Ruin

“…Albrecher and Boxma [9] study the expected discounted penalty function in a semi-Markovian dependent risk model in which at each instant of a claim, the underlying Markov chain jumps to a new state and the distribution of claim depends on this state. Liu et al [10,11] consider the expected discounted penalty function under the constant dividend barrier and dividends payments under the threshold strategy in a Markov-dependent risk model, respectively.…”

“…Note that given the states 1 n Z  and n Z , the quantities n and n W X are independent, but there is an autocorrelation among consecutive claim sizes and among consecutive interclaim times as well as crosscorrelation between and n n W X . Inspired by Albrecher and Boxma [9] and Liu et al [10,11], in this paper we propose to generalize the semiMarkovian risk model to the absolute ruin risk model. In the new risk model, we assume that the insurer could borrow an amount of money equal to the deficit at a debit interest force  when the surplus is negative.…”