Mathematical Problems in Engineering

Volume 2015, Article ID 874101, 13 pages

http://dx.doi.org/10.1155/2015/874101

## Homogeneous Discrete Time Alternating Compound Renewal Process: A Disability Insurance Application

^{1}Dipartimento di Farmacia, Università “G. d’Annunzio” di Chieti, Via dei Vestini 31, 66013 Chieti, Italy^{2}Guglielmo Marconi University, Via Plinio 44, 00193 Roma, Italy^{3}Solvay Business School, Universitè Libre de Bruxelles, Avenue Franklin D. Roosevelt, CP 145/1-21, 1050 Brussels, Belgium^{4}Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza, Università di Roma “La Sapienza”, Via del Castro Laurenziano 9, 00161 Roma, Italy

Received 19 March 2015; Revised 1 June 2015; Accepted 11 June 2015

Academic Editor: Babak Shotorban

Copyright © 2015 Guglielmo D’Amico et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Discrete time alternating renewal process is a very simple tool that permits solving many real life problems. This paper, after the presentation of this tool, introduces the compound environment in the alternating process giving a systematization to this important tool. The claim costs for a temporary disability insurance contract are presented. The algorithm and an example of application are also provided.

#### 1. Introduction

It is possible to assert that the roots of renewal processes are in actuarial science. Indeed, the seminal paper Lundberg [1] introduced this theory and was written by an actuary. Other two seminal papers [2, 3] were written by a really important probability researcher that did many applications in insurance.

For a complete introduction to the renewal processes, we refer to the book Cox [4].

As well known, renewal processes work in this way. We have a phenomenon that will be verified but we do not know when. When the studied phenomenon happens, then the system is renewed and, in homogeneous case, it restarts with the same initial characteristics (for first results in nonhomogeneous renewal processes, see Gismondi et al. [5]). It is clear that a simple actuarial model can be well simulated by this kind of stochastic process.

Suppose that we have to study a system that can assume two different values that cannot be verified together and that when one of them is not verified then the other is verified and vice versa. In this case, the consideration of only the renewal is not sufficient for the study of the problem. The random evolution of these processes can be studied by means of the alternating renewal processes that are a generalization of the renewal process (see Figure 6). These processes were applied in many fields (see, e.g., Zacks [6], Di Crescenzo et al. [7], and Elsayed [8]).

The renewal processes had and have a great relevance in Actuarial Sciences, but there were few applications of the alternating renewal processes.

In health insurance models, there are two kinds of randomness: one is given by the state of the insured (healthy or ill) and the other by the duration inside the states. This phenomenon can be well simulated by the alternating renewal processes. Both the transitions between the two states are possible. An ordinary renewal process cannot model this insurance contract. Ramsay [9] and more recently Adekambi and Mamane [10] proposed the alternating renewal process for the study of health insurance problems. The used tool fits well in the problem. However, the continuous time environment used in both the papers is really difficult to apply. Indeed, renewal processes in most cases should be solved numerically. For example, Adekambi and Mamane [10] constructed an interesting model in which they derive the first two moments of the aggregate claim amount of benefit paid out up to a given time . The problem is that although the mathematical apparatus is adequate, the application was done in a very simple case with negative exponential probability distribution functions. D’Amico et al. [11] showed how the application of discrete time alternating renewal processes is simple.

For a general reference on the discrete time renewal processes, we recall Feller [12].

This paper will generalize the results presented in D’Amico et al. [11] applying the alternating compound renewal process for the study of insurance contracts in a discrete time environment.

Discrete time alternating compound renewal processes were described in Tijms [13] and in Beichelt [14]. In both of these books, the alternating compound processes were introduced as exercises. In Beichelt, the exercise was only described but not solved and no hints were given for solving it. In the Tijms book, an asymptotic solution was proposed not a solution that could give the time evolution for a given time interval. We think that, in general, the applications should present the time evolution of the studied phenomenon. In particular, in the study of a mechanical system where usually the lifetime of the studied apparatus is shorter than 20, or at most 30 years, the study of the evolution in time of the mechanical system should be by far more important. Furthermore, in the chapter on advanced renewal theory of the Tijms book at page 326, the following is written:

“in many applied probability problems, asymptotic expansions provide a simple alternative to computationally intractable solutions.”

We would outline that, in this paper, it is shown that the numerical solution of alternating compound renewal can be obtained in a simple way although the process is a strong generalization of the simple renewal process.

Only the paper Zacks [6] presented an application of alternating compound processes in telegraph problem, but in the particular framework of the Poisson processes not in the general environment of the renewal processes.

In Alvarez [15], a theoretical paper on the alternating renewal processes, in which how to solve the evolution equation analytically but always in a Poisson environment is shown, was presented. The possibility to apply to an engineering problem this tool was also described. But without it any example of this application was not given.

Another paper, Vlasiou [16], gives a very short presentation of alternating compound processes asserting that the total rewards earned are equal to the percentage of the time spent in the UP state with respect to the total time . This result holds in very particular cases and not in a general framework where the rewards, for example, are given by sum of money.

Furthermore, to authors’ knowledge, and in any database consulted, never was a paper presented where the costs and the revenues for each period of the time horizon were calculated.

In the previous paper (D’Amico et al. [11]), the authors proposed the application of a discrete time of the simple alternating renewal process in disability insurance, where the rewards were not considered.

The main purpose of this paper is to show how it is possible to apply discrete time alternating compound renewal processes in insurance problems. More specifically, the application will be presented in temporary disability insurance problem generalizing the previous results.

The obtained results are general and can be applied in any other field.

The paper will develop in the following way. In the second section, the discrete time alternating processes in a homogeneous environment will be presented recalling some results obtained in D’Amico et al. [11]. Furthermore, in that paper, the discrete time approach was justified by discussing the strict relation between continuous and discrete time alternating renewal processes. This relation was proved adapting the results obtained in Corradi et al. [17] for homogeneous semi-Markov processes.

In Section 3, the discrete time compound renewal processes will be reported.

In Section 4, the discrete time alternating compound renewal model for the calculation of the mean values (MV), in nondiscounted case, and the mean present values (MPV), in discounted case, is presented. In this way, all the total rewards paid by the insurers for the premiums and by the insurance company for the reported claims in temporary disability insurance will be calculated. In Section 5, examples of the application of the model will be presented. In the last section, some concluding remark will be given.

In this paper, we will follow the notation given in Beichelt [14].

#### 2. Discrete Time Homogeneous Alternating Processes

In renewal theory, usually, it is supposed that renewals start as soon as they happen. In real world, it is possible that this condition is not satisfied; that is, renewals can start after a nonnegligible random time. It is possible to take into account this phenomenon, defining a renewal process in which the renewal time after the failure is assumed being an integer nonnegative random variable.

*Definition 1. *Let and be two random variables (one will denote r.vs. in this way in singular or plural case) that are supposed to be two independent sequences of independent nonnegative r.v. In this way, the sequence of two-dimensional random vectors is defined to be a discrete time alternating renewal process. denotes the th working period and denotes the th nonworking period.

It is posed that at time 0^{+} the system is working and this is the reason why precedes .

*Remark 2. *It is important to outline that the working state depends on the application. In a reliability of an engineering system, then UP state corresponds to the working of the system. In a temporary disability insurance contract, then for the insurance, the UP state is the absence of disability.

The two integer random variables,represent the times at which the failures happen and the renewed system starts working, respectively.

From (1), it results inNow, it is possible to define a state system indicator variable, more precisely,In Figure 1, a trajectory of the r.v. (3) is reported.