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

Abstract

We consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a time-changed Brownian motion. The time-change is obtained via the inverse of an α-stable subordinator. These processes are specified in terms of time-changed Gauss-Markov processes and fractional time-changed diffusions. A fractional pseudo-Fokker-Planck equation for such processes is given. We investigate their first passage time densities providing a generalized integral equation they satisfy and some transformation rules. First passage time densities for time-changed Brownian motion and Ornstein-Uhlenbeck processes are provided in several forms. Connections with closed form results and numerical evaluations through the level zero are given.

Keywords:

• First passage time
• fractional diffusion
• time-changed diffusion
• integral equation
• numerical evaluation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 60G22
• 26A33

Acknowledgements

The authors are thankful to the referees for careful reading of the manuscript and many detailed comments and suggestions that helped to improve it.

Notes

1 Note that ${\zeta }_D(t)\equiv \zeta (t,t)$ of Proposition 2.1.

Additional information

Funding

This work was partially supported by MIUR—PRIN 2017, project Stochastic Models for Complex Systems, no. 2017JFFHSH and by Gruppo Nazionale per il Calcolo Scientifico (GNCS-INdAM).