Ruin problems for a discrete time risk model with non-homogeneous conditions

Abstract

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

Bi-seasonal discrete time risk model

“…Theorem 1.1. Consider the inhomogeneous discrete time risk model defined by equality (1.1), in which u 2 N 0 and According to this theorem, we can calculate the finite time ruin probability wðu; TÞ ¼ w ð0Þ ðu; TÞ of the initial model for all u 2 N 0 and T 2 N. Similar formulas for the finite time probabilities and related quantities can be found in [2,4,29,30].…”

On the ruin probability for nonhomogeneous claims and arbitrary inter-claim revenues

“…However, such a non-i.d. assumption generates serious difficulties in the ruin probability evaluation, hence the related papers generally discuss recursions for the finite time ruin probability for discrete-time risk processes under restrictive assumptions, like: De Kok [15], who presented an approximate recursive algorithm based on the first two moments of the CSs distributions; Blazevicius et al [16] obtained recursions for discrete and rational-valued CSs; or Castaner et al [17], whose recursive algorithm needs the discretization of the CSs distributions.…”

An Analysis of Black-Box Optimization Problems in Reinsurance: Evolutionary-Based Approaches

“…It is necessary to consider the probability of survival of the insurer with an stop-loss (ϕ I (c R )), that can be obtained adapting the univariate model explained in [2]. We must also consider the probability of survival of the reinsurer given the insurer's survival.…”