
Date: Thursday, March 21 
Session: 78 

ASTIN 


Jacques Janssen
curriculum
, Raimondo Manca
curriculum

Belgium, Italy 

Subject 

Session 





Paper 



General Actuarial Models in a SemiMarkov Environment 


Presentation 



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Summary
The first application of SemiMarkov 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 SemiMarkov Reward Processes (HSMRP). The Discrete Time NonHomogeneous SemiMarkov 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 nonhomogeneous semiMarkov processes. It is to outline that the models that are obtained for all actuqarial applications that the author constructed are similar. They bring to SMRP in which it is possible to sonsider simultaneousely 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 semiMarkov 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 semiMarkov 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 semiMarkov 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 wighted 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 simultaneusely all the aspects of a general actuarial problem will be given in the paper. That formula is able to take in account all the possibillity 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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Jacques Janssen 

Curriculum

Professeur et docteur en sciences et licencié en sciences actuarielles de l’Université Libre de Bruxelles, est actuellement Président du CESIAF (Centre d’Etudes Interuniversitaire d’Assurfinance) et professeur à à l’UBO pour l’EURIA (EuroInstitut d’Actuariat Jean Dieudonné, Brest) et Président de l’ITA (Institut des Techniques Actuarielles, France).
Ses recherches actuelles traitent de la Modélisation stochastique appliquée en finance, assurance, gestion et économie.
Il est membre de l’ARAB, de l’Association des Actuaires Suisses, de l’AAI ainsi que de l’ISI (International Statistical Institute) et auteur de nombreux livres et articles scientifiques.


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Prof. Raimondo Manca short C.V. 

Curriculum

Position: Full professor in Mathematics for the applications in Economics, Finance and Actuarial Science at University “La Sapienza” Rome.
Topics of research: Application of Stochastic processes to Finance and Actuarial Science, Algorithms in Actuarial Science and Finance, Computational Probability, Data Structures, Linear Algebra, Sparse Matrices.
Member of IAA, ASTIN, AFIR/ERM, IIA, AMASES. He wrote many scientific articles on his research topics.


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