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Bernstein Fractional Derivatives: Censoring and Stochastic Processes

David Berger, Cailing Li, René L. Schilling

TL;DR

This work develops a unified analytic-probabilistic framework for censored fractional Bernstein derivatives on the half-line. It defines the Bernstein fractional derivative $_0^{ m R}{ }^{f}D$ and its censored version $_0^{ m Ce}{ }^{f}D$, derives explicit inverses via the Bernstein integral $_0^{ m R}{ }^{f}I$ and a convergent kernel-series, and solves censored initial-value and resolvent problems. The paper also constructs the censored decreasing subordinator $S^c$ by a piecing-out procedure, proves that $S^c$ is a Feller process with lifetime finite almost surely, and provides detailed probabilistic representations linking the semigroup and resolvent to the censored Bernstein derivative. By embedding the results in the Bernstein functional calculus and Sonine-pair machinery, it extends censored fractional calculus on the half-line to a robust stochastic-process setting with potential applications in physics and biology.

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 $S^c$.

Bernstein Fractional Derivatives: Censoring and Stochastic Processes

TL;DR

This work develops a unified analytic-probabilistic framework for censored fractional Bernstein derivatives on the half-line. It defines the Bernstein fractional derivative and its censored version , derives explicit inverses via the Bernstein integral and a convergent kernel-series, and solves censored initial-value and resolvent problems. The paper also constructs the censored decreasing subordinator by a piecing-out procedure, proves that is a Feller process with lifetime finite almost surely, and provides detailed probabilistic representations linking the semigroup and resolvent to the censored Bernstein derivative. By embedding the results in the Bernstein functional calculus and Sonine-pair machinery, it extends censored fractional calculus on the half-line to a robust stochastic-process setting with potential applications in physics and biology.

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 , which is obtained either via a pathwise construction by removing those jumps from the decreasing subordinator , , 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 .

Paper Structure

This paper contains 7 sections, 23 theorems, 157 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Bernstein derivatives
  3. Censored initial value problem
  4. Resolvent equation
  5. Construction of the censored process
  6. Probabilistic representation
  7. Positive Sonine pairs and Bernstein functions

Key Result

Lemma 2.2

Let $f$ be a Bernstein function with triplet $(0,0,\mu)$, $f_n = nf/(n+f)$ as in bd-e28, and $\phi,\psi\in L^1(0,\infty)$. The following assertions are equivalent

Figures (1)

  • Figure 1: The processes ${}^xS^c_t - {}^yS^c_t$ "march" in parallel as long as they are not or always simultaneously censored. This changes at the first censoring event, where only one of the processes is censored. This situation is shown for $\tau_1$, where only ${}^yS^c$ is censored, hence the distance ${}^xS^c_{\tau_1} - {}^yS^c_{\tau_1}<x-y$.

Theorems & Definitions (56)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 46 more