First passage times estimates for time-changed Brownian motion: paths simulation and hazard-rate methods

Anomalous diffusions are often observed in real data, that can manifest in asymmetric densities, heavy tails, sharp peaks, different spreading rate.Among such diffusions, we can find some time-changed processes, obtained by composing two independent processes: the outer process and the inverse of an α-stable subordinator. The central role is played by the time-changed Brownian motion (TCBM), that is characterized by continuous sample paths having time periods (of a random duration) in which the process remains trapped. Sometimes this process is also called delayed Brownian motion (BM). For application purposes the first passage time (FPT) of this process is the core of many studies but no closed form results are known. We refer to a subordinated density function of the FPT of the time-changed Brownian motion as the integral of the FPT density of the BM with respect to the density of the inverse stable subordinator. Taking this function into account, our idea is to generalize the hazard rate method (HRM) that we implemented in an R package for FPT simulation of Gaussian diffusions. We propose two different generalizations of the HRM by considering subordinated hazard rates. Results are provided in graphical form for different values of the stability order of the subordinator in case of a constant threshold. Comparisons are also made with FPT obtained by simulation of the trajectories of the TCBM, in order to verify the agreement of the two approaches. Join work with Maria Francesca Canfora (Consiglio Nazionale delle Ricerche) and Enrica Pirozzi (Università della Campania Luigi Vanvitelli).