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
In this paper we present a threshold proportional reinsurance strategy and we analyze the effect on some solvency measures: ruin probability and time of ruin. This dynamic reinsurance strategy assumes a retention level that is not constant and depends on the level of the surplus. In a model with inter-occurrence times generalized Erlang(n)-distributed we obtain the integro-differential equation for the Gerber-Shiu function. Then, we present the solution for inter-occurrence times exponentially distributed and claim amount phase-type(N). Some examples for exponential and phase-type(2) claim amount are presented. Finally, we show some comparisons between threshold reinsurance and proportional reinsurance.
Classical intervals have been a very useful tool to analyze uncertain and imprecise models, in spite of operative and interpretative shortcomings. The recent introduction of modal intervals helps to overcome those limitations. In this paper, we apply modal intervals to the field of probability, including properties and axioms that form a theoretical framework applied to the Markovian analysis of Bonus-Malus systems in car insurance. We assume that the number of claims is a Poisson distribution and in order to include uncertainty in the model, the claim frequency is defined as a modal interval; therefore, the transition probabilities are modal interval probabilities. Finally, the model is exemplified through application to two different types of Bonus-Malus systems, and the attainment of uncertain long-run premiums expressed as modal intervals.
This paper is focused on solving different hard optimization problems that arise in the field of insurance and, more specifically, in reinsurance problems. In this area, the complexity of the models and assumptions considered in the definition of the reinsurance rules and conditions produces hard black-box optimization problems (problems in which the objective function does not have an algebraic expression, but it is the output of a system (usually a computer program)), which must be solved in order to obtain the optimal output of the reinsurance. The application of traditional optimization approaches is not possible in this kind of mathematical problem, so new computational paradigms must be applied to solve these problems. In this paper, we show the performance of two evolutionary and swarm intelligence techniques (evolutionary programming and particle swarm optimization). We provide an analysis in three black-box optimization problems in reinsurance, where the proposed approaches exhibit an excellent behavior, finding the optimal solution within a fraction of the computational cost used by inspection or enumeration methods.
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