Stochastic processes governed by the generalized telegraph process and by Brownian motion and their applications

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]