MARKOV PROCESSES IN FINANCE AND INSURANCE

In this thesis we tried to make one more step in the application of Markov processes in the actuarial and financial field. Two main problems have been dealt. The first one regarded the application of a Markov process for the description of the salary lines of participants in an Italian Pension Scheme of the First Pillar. A semi-Markov process with backward recurrence time was proposed. A statistic test has been applied in order to determine whether the null hypothesis of a geometrical distribution of the waiting times of the process should be accepted or not. The test showed that the null hypotheses was rejected for some of the waiting time distributions and thus we concluded that the semi-Markov process should be preferred to the simple Markov chain to model the transition in the states of the salary process. In the financial application, we treated the Indexed semi-Markov chain, a new model that has been previously used to describe intra-day price return dynamics. The peculiarity of this model is that, through the Index process, it manages two very known stylized facts of financial time series: the first one is the long memory of financial series and the second one is the volatility clustering. This is achieved by defining the Index as a function of the m-th previous values of the price returns. In order to transform the values obtained in states of a stochastic process a discretization of the Index is necessary. We proposed the method of change points as a new method to obtain the most efficient classes. This approach is justified by the fact that, for financial time series, the price dynamics in different levels of volatility in the market present different characteristics. We found out that the best discretization of the Index process is that of using four change points, which implied five levels of volatility in the market: very low, medium low, medium, medium high and very high. We also generated synthetic trajectories in order to calculate the autocorrelation of the square of returns for the real data as well as for the hypothesized models. The autocorrelation function showed that the model with four change points was the closest to the real data.