| dc.description.abstract | The aim of this research thesis is to broaden the treatment of stochastic
processes derived from the telegraph process and Brownian motion. First,
the classical telegraph process is presented. The Kolmogorov equations and
the telegraph equation for the transition density are derived, as well as an explicit
form for the latter. The results are extended to a generalization of the
telegraph process that admits velocities with different absolute value, whose
alternation is governed by a generic counting process with independent increments.
Then, a generalization of the telegraph process whose sample-paths
fluctuates around the zero state is considered. The latter process describes
the motion of a particle on the real line, which is characterized by constant
velocities and state-dependent intensities that vanish when the motion is toward
the origin. This assumption allows to adopt an approach based on
renewal theory to obtain formal expressions of the forward and backward
transition densities of the process. The special case when certain random
times of the motion possess gamma distribution leads to closed-form expressions
of the transition densities, given in terms of the generalized Mittag-
Leffler function. A first-passage-time problem for the considered process is
also addressed. Then a stochastic process defined as the sum of a Brownian
motion and a generalized integrated telegraph process is presented, focusing
in particular on the case in which the inter-arrival times have an exponential
distribution. Explicit forms are derived for the transition density and flow
function of the process, as well as a differential system that generalizes the
Kolmogorov equations. Based on the latter process, a stochastic model to describe
the vertical motions in the Campi Flegrei volcanic region is proposed.
After a brief description of the phenomenon and of the available data sets,
a statistical procedure is presented to identify the points of changes in the
trend of the motion and the corresponding inflation and deflation episodes,
as well as to estimate the velocities of motion, the turning rates and the
infinitesimal variance of motion. The estimates obtained, combined with the
knowledge of the probability laws that regulate the motion, enables predictions
of ground displacements and their changing tendency at future time
instants. To verify the admissibility of the model, a statistical test on the
Brownian component is performed. Finally, there is room for some remarks
on the obtained results and considerations on possible future developments. [edited by Author] | it_IT |
