Censored fractional Bernstein derivatives and stochastic processes

TL;DR

This paper introduces a new censored fractional Bernstein derivative linked to Bernstein Riemann–Liouville derivatives, demonstrating its role as a generator of censored subordinators and analyzing boundary hitting times.

Contribution

It defines a novel censored fractional Bernstein derivative and establishes its connection to censored subordinators and boundary behavior.

Findings

The censored fractional Bernstein derivative is the generator of censored subordinators.

Censored subordinators hit the boundary in finite time under specific conditions.

The derivative is constructed via solving a resolvent equation.

Abstract

In this paper, we define the censored fractional Bernstein derivative on the positive half line $(0, \infty)$ based on the Bernstein Riemann--Liouville fractional derivative. This derivative can be shown to be the generator of the censored subordinator by solving a resolvent equation. We also show that the censored subordinator hits the boundary in finite time under certain conditions.