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Stochastic Processes and Mean Square Calculus on Fractal Curves

Alireza Khalili Golmankhaneh, Kerri Welch, Cristina Serpa, Ivanka Stamova

Abstract

In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence of random variable on fractal curve, fractal mean square continuity, mean square $F^α$-derivative, and fractal mean square integral. The mean square solution of a fractal stochastic equation is derived and plotted in order to show the details.

Stochastic Processes and Mean Square Calculus on Fractal Curves

Abstract

In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence of random variable on fractal curve, fractal mean square continuity, mean square -derivative, and fractal mean square integral. The mean square solution of a fractal stochastic equation is derived and plotted in order to show the details.
Paper Structure (7 sections, 11 theorems, 99 equations, 2 figures)

This paper contains 7 sections, 11 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractal calculus on fractal curve
  4. Random variable on fractal curve
  5. Random processes on fractal curve
  6. Mean square $F^{\alpha}$-calculus on fractal curves
  7. Conclusion

Key Result

Theorem 4.1

Let $X_{n}$ and $X_{n'}'$ where $n,n'\in N ~(=index ~set)$ be two sequence of second order f.r.v., if where $n_{0}$ and $n_{0}'$ are limit points of $N$. Then

Figures (2)

  • Figure 1: Graph of Eq. \ref{['EDEES']} choosing $\lambda=1$
  • Figure 2: Graph of Eq. \ref{['ESwqa']} choosing $E[X_{1}]=0, E[X_{0}]=1, E[A^2]=4$

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 51 more