Session: 78

ASTIN

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Summary

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 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 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 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.

Session: 47

AFIR/ERM

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Summary

Homogeneous semi-Markov processes (HSMP) were defined in the fifties (Levy 1954). At beginning their applications were in engineering field, mainly in problem of reliability and maintenance (see Howard 1972). In general this processes are a tool that is useful to study problems linked to the ageing of a system.

Janssen (1966) applied for the first time HSMP in actuarial science. In this field there were many applications of HSMP. Also in this field SMP can be considered a tool that gives good results in real world applications. 

Non Homogeneous Semi-Markov Processes (NHSMP) were introduced by Iosifescu Manu (1972) ; these processes were applied in actuarial field see (Hoem 1972, CMIR 1991) and more recently (Haberman, Pitacco 1999, Pitacco, Olivieri 1997).

The Discrete Time Non-Homogeneous Semi-Markov Process (DTNHSMP) were introduced in De Dominicis, Janssen (1984) and a real data actuarial application was presented in this paper that was later on generalized and applied to actuarial problem, see for example (Janssen, Manca 1998).

It is to outline that homogeneous and non-homogeneous SMP are a generalization of the corresponding Markov processes (MP) in the sense that in MP environment the waiting time distribution function (d.f.) in a state before a change is exponential whereas in SMP can be of any type.

In the last twenty years finance made great theoretical advances using stochastic tools starting from the two fundamentals papers respectively of (Black & Scholes 1973) on option pricing and of (Vasicek 1977) on stochastic interest rate. Innovative results were obtained in all financial topics introducing massively probability tools. In some cases the results generalized the ones given in deterministic environment, for example the ones from (Buhlmann 1992), (Buhlmann 1994) and (Norberg 1995) on the evaluation of stochastic interest rate. In other cases the probabilistic approach gave the possibility to evaluate some financial tools that otherwise should be really difficult to study in a realistic way (see Willmott 1998), mainly for derivative products. Furthermore the opportunity to face in a more correct way some financial tools allowed the introduction of new products, like some special kind of options (see Willmott 1998). 

The main part of these results were obtained in homogeneous MP environment. It is to outline that, in authors opinion, the initial time of a financial operation is usually known, whereas the time when the operation will be closed is not known, furthermore changing the initial time of a financial operation usually changes the conditions (as well known this fact is taken in account by means of non homogeneity). This observations imply that it could be really interesting to introduce a stochastic environment also for the time length of financial operations considering also the initial time of operations. This step can be made using NHSMP models. This step was already proposed in some theoretical papers (Svishchuk 1995, Svishchuk, Burdeinyi 1996, Janssen, Manca, De Medici 1996, Janssen, Manca, Volpe di Prignano 1999, Janssen, Manca, Di Biase 1997, 2001) in homogeneous case and (Janssen, Manca 2000, Janssen, Manca, Di Biase 1998) in the non-homogeneous one. 

A model useful to construct term structure of implied forward rates in a stochastic environment will be presented in this paper. The stochastic process used to construct the interest rate structure will be the NHSMP, in the paper the financial operations are supposed non-homogeneous in time (see Volpe di Prignano 1985).

It is to precise that a discrete time framework will be used in the paper. There are two reasons to apply discrete time non-homogeneous semi-Markov process (DTNHSMP) instead that the continuous one. The first is that in a previous paper (Janssen, Manca 2001) was proved that discretising the evolution equation of a continuous time NHSMP by means of the simplest quadrature formula the DTNHSMP is obtained and if the time interval of DTNHSMP tends to 0 then is obtained the continuous case. The second is that the to solve DTNHSMP evolution equation is not a difficult task, as the solution of the continuous time can be obtained analytically only in same special cases; in the other cases to find the solution it is necessary to work numerically and the most simple way to obtain the numerical solution of the process evolution equation is the one by which the DTNHSMP is obtained by the continuous time NHSMP. 

In the second paragraph will be introduced the theoretical stochastic interest rate model, in the third one will be presented the DTNHSMP in the forth paragraph will be explained how to implement the model by means of DTNHSMP and in the last part will be given an applicative example.

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