This paper is concerned with a non-homogeneous discrete time risk model where premiums are fixed but non-uniform, and claim amounts are independent but non-stationary. It allows one to account for the influence of inflation and interest and the effect of variability in the claims. Our main purpose is to develop an algorithm for calculating the finite time ruin probabilities and the associated ruin severity distributions. The ruin probabilities are shown to rely on an underlying algebraic structure of Appell type. That property makes the computational method proposed quite simple and efficient. Its application is illustrated through some numerical examples of ruin problems. The well-known Lundberg bound for ultimate ruin probabilities is also reexamined within such a non-homogeneous framework.Keywords: Discrete time risk model; Non-uniform premiums; Non-stationary claims; Rates of interest; Finite time ruin probability; Ruin severity distribution; Computational methods; Lundberg bound 1 A non-homogeneous discrete time risk modelThe classical compound Poisson and binomial risk models assume that the premium income is constant over time and the claim amounts form a sequence of independent identically distributed (i.i.d.) random variables. These assumptions of homogeneity of premiums and claim amounts can be too restrictive in reality, especially because of the influence of the economic environment. For instance, inflation and interest can affect, sometimes drastically, the evolution of the reserves of the company. Claim amounts and premiums have often a tendency to increase for various socio-economic reasons (e.g. higher loss levels and larger compensations or coverages).In this section, we generalize the compound binomial risk model in order to account for such factors of non-homogeneity. For that, it will be necessary to specify, inter alia, the time when premiums are collected and how they are evaluated. To begin with, we are going to consider a particular model which incorporates arbitrary fixed interest rates.The influence of interest force. Risk theory with interest income has received an increasing attention in the literature. A number of works are devoted to models

This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

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This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

No abstract