Bernstein Fractional Derivatives: Censoring and Stochastic Processes

TL;DR

This paper introduces censored fractional Bernstein derivatives on the positive half-line, linking them to censored decreasing subordinators and their probability distributions, with implications for stochastic process analysis.

Contribution

It defines a new class of censored fractional Bernstein derivatives and establishes their connection to censored decreasing subordinators and their distributions.

Findings

Censored fractional Bernstein derivatives are generators of censored decreasing subordinators.

The censored subordinator has finite lifetime.

Various probability distributions related to the censored subordinator are identified.

Abstract

We define censored fractional Bernstein derivatives on the positive half-line based on the Bernstein--Riemann--Liouville fractional derivative. The censored fractional derivative turns out to be the generator of the censored decreasing subordinator $S^c = (S_t^c)_{t\geq 0}$, which is obtained either via a pathwise construction by removing those jumps from the decreasing subordinator $(x-S_t)_{t\geq 0}$, $x>0$, that drive the path into negative territory, or via the Hille--Yosida theorem. Then we show that the censored decreasing subordinator has only finite life-time, and we identify various probability distributions related to $S^c$.