General Actuarial Models in a Semi-Markov Environment

The first application of Semi-Markov Process (SMP) in actuarial field was given by J.Janssen [6]. Many authors successively used these processes and their generalizations for actuarial applications, (see Hoem, [4], Carravetta, De Dominicis, Manca, [1], Sahin, Balcer, [11]. In some books it is also shown how it is possible to use these processes in actuarial science, (see Pitacco, Olivieri, [10], CMIR12 [12]. These processes can be generalised introducing a reward structure see for example Howard, [5], in this way are defined the Homogeneous Semi-Markov Reward Processes (HSMRP). The Discrete Time Non-Homogeneous Semi-Markov Reward Processes (DTNHSMR) were introduced in De Dominicis, Manca [2]. At the author knowing these processes in actuarial field were introduced only for the construction of theoretical models that were not yet applied. (see De Dominicis, Manca, Granata, [3]. Janssen, Manca, [8],[9]. The applications proposed in those papers were in pension and in health insurance. The author is also working on the construction of a model for non life insurance, more precisely on motor car liabilities, using non-homogeneous semi-Markov processes. It is to outline that the models that are obtained for all actuarial applications that the author constructed are similar. They bring to SMRP in which it is possible to consider simultaneously the future development of the state system and its financial evolution. The figure 1 is reported from Pitacco, Olivieri, [10]. The two authors explain that this can be considered a graph that gives a trajectory of the stochastic process that describes an insurance operation. The figure 2 gives the trajectory of a possible evolution of a semi-Markov process (see Janssen, [7]. It is evident that they have the same behaviour. And this can explain because the actuarial models, in the author opinion, are strictly connected to semi-Markov processes. In this light and after many experiences, the author think that it is possible to face any kind of actuarial problem by means of a model based on SMP. In this paper a semi-Markov reward model that can afford a general actuarial problem will be presented. The graph describing this model is reported in figure 3. It is to precise that the arcs are weighted and their weights can represent the change state probabilities and the rewards that are paid in the case of change state Furthermore, also the nodes, that represent the model states, are weighted and their weights represent the reward paid or received remaining in the state. All rewards can be fixed or can change in the time evolution of the model. The formula of the evolution equation of a SMRP that can take in account simultaneously all the aspects of a general actuarial problem will be given in the paper. That formula is able to take in account all the possibility that can happen in an actuarial problem. In the paper will be explained how the formula and the related graph will change in function of the actuarial problem that is to face.

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