A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Abstract

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

First passage times for some classes of fractional time-changed diffusions

“…An advanced investigation about other possible transformed closed forms such as those related with one-side and two-side Daniels-type boundaries ( [17], [35]), but also additional asymptotic results ( [7], [18]) will be the object of a future work.…”

“…In Section 3 the pseudo Fokker-Planck equation is proved to hold for a class of fractional time-changed diffusions related to the above processes. In Section 4 the FPT topic is addressed: an integral equation is provided following the Volterra integral approach ( [17], [35]) specialized for the specified time-changed processes. Due its key rule, we focus on FPT density of the time-changed Brownian motion.…”