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Sibuya probability distributions and numerical evaluation of fractional-order operators

Nikolai Leonenko, Igor Podlubny

TL;DR

The paper addresses numerical evaluation of fractional-order operators via Monte Carlo methods using the $Sibuya$ distribution. It develops three simulation methods for the Sibuya distribution, introduces Sibuya-like distributions and a continuous Sibuya distribution for Riemann–Liouville integration, and provides a general MC framework for signed nonlocal operators. The contributions include explicit MC schemes for $0<\alpha<1$, structured extensions to higher orders with signed probabilities, sieved variants for $2<\alpha<3$ and beyond, and MATLAB toolboxes with illustrative examples. This work enhances numerical fractional calculus applications by enabling efficient MC-based differentiation and integration with potential for broad stochastic modeling and applications.

Abstract

In this work we explore the Sibuya discrete probability distribution, which serves as the basis and the main instrument for numerical simulations of Grunwald--Letnikov fractional derivatives by the Monte Carlo method. We provide three methods for simulating the Sibuya distribution. We also introduce the Sibuya-like sieved probability distributions, and apply them to numerical fractional-order differentiation. Additionally, we use the Monte Carlo method for evaluating fractional-order integrals, and suggest the notion of the continuous Sibuya probability distribution. The developed methods and tools are illustrated by examples of computation. We provide the MATLAB toolboxes for simulation of the Sibuya probability distribution, and for the numerical examples.

Sibuya probability distributions and numerical evaluation of fractional-order operators

TL;DR

The paper addresses numerical evaluation of fractional-order operators via Monte Carlo methods using the distribution. It develops three simulation methods for the Sibuya distribution, introduces Sibuya-like distributions and a continuous Sibuya distribution for Riemann–Liouville integration, and provides a general MC framework for signed nonlocal operators. The contributions include explicit MC schemes for , structured extensions to higher orders with signed probabilities, sieved variants for and beyond, and MATLAB toolboxes with illustrative examples. This work enhances numerical fractional calculus applications by enabling efficient MC-based differentiation and integration with potential for broad stochastic modeling and applications.

Abstract

In this work we explore the Sibuya discrete probability distribution, which serves as the basis and the main instrument for numerical simulations of Grunwald--Letnikov fractional derivatives by the Monte Carlo method. We provide three methods for simulating the Sibuya distribution. We also introduce the Sibuya-like sieved probability distributions, and apply them to numerical fractional-order differentiation. Additionally, we use the Monte Carlo method for evaluating fractional-order integrals, and suggest the notion of the continuous Sibuya probability distribution. The developed methods and tools are illustrated by examples of computation. We provide the MATLAB toolboxes for simulation of the Sibuya probability distribution, and for the numerical examples.
Paper Structure (25 sections, 1 theorem, 91 equations, 13 figures)

This paper contains 25 sections, 1 theorem, 91 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Grünwald--Letnikov fractional derivative
  3. Sibuya distribution
  4. Method 1
  5. Method 2
  6. Method 3
  7. Numerical simulation of Sibuya distribution
  8. Sibuya-like distributions and non-local operators
  9. The general Monte Carlo procedure
  10. The case of $0 < \alpha < 1$
  11. The case of $1 < \alpha < 2$
  12. The case of $2 < \alpha < 3$
  13. The case of $3 < \alpha < 4$
  14. The case of $4 < \alpha < 5$
  15. The Monte Carlo method for numerical fractional integration using continuos Sibuya-like distibution
  16. ...and 10 more sections

Key Result

Theorem 1

Let $B$ has a beta distribution $B(\alpha, 1-\alpha)$. Further, conditionally $B = p$, which has a geometric distribution (geom-distribution) Then, unconditionally, $Y$ has a Sibuya distribution with parameter $\alpha \in (0, 1)$.

Figures (13)

  • Figure 1: Simulation of the Sibuya distribution by methods 1 (top), 2 (middle), and 3 (bottom) for (left to right) $\alpha=0.1$, $\alpha=0.3$, $\alpha=0.5$, $\alpha=0.7$, $\alpha=0.9$, with 100 draws of 1000 numbers each.
  • Figure 2: Comparison of the methods 1, 2, and 3 using the sorted averages of the the drawn random numbers for $\alpha=0.3$ (left), $\alpha=0.5$ (middle), $\alpha=0.7$ (right).
  • Figure 3: The values of $w_k$, $k = 1, 2, \ldots$, in case $0<\alpha<1$, with $\alpha = 0.5$
  • Figure 4: The values of $w_k$, $k = 1, 2, \ldots$, in case $1<\alpha<2$, with $\alpha = 1.5$
  • Figure 5: The values of $w_k$, $k = 1, 2, \ldots$, in case $2<\alpha<3$, with $\alpha = 2.5$
  • ...and 8 more figures

Theorems & Definitions (1)

  • Theorem 1