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Entropy of Wiener integrals with respect to fractional Brownian motion

Iryna Bodnarchuk, Yuliya Mishura, Kostiantyn Ralchenko

Abstract

The paper is devoted to the properties of the entropy of the exponent-Wiener-integral fractional Gaussian process (EWIFG-process), that is a Wiener integral of the exponent with respect to fractional Brownian motion. Unlike fractional Brownian motion, whose entropy has very simple monotonicity properties in Hurst index, the behavior of the entropy of EWIFG-process is much more involved and depends on the moment of time. We consider these properties of monotonicity in great detail.

Entropy of Wiener integrals with respect to fractional Brownian motion

Abstract

The paper is devoted to the properties of the entropy of the exponent-Wiener-integral fractional Gaussian process (EWIFG-process), that is a Wiener integral of the exponent with respect to fractional Brownian motion. Unlike fractional Brownian motion, whose entropy has very simple monotonicity properties in Hurst index, the behavior of the entropy of EWIFG-process is much more involved and depends on the moment of time. We consider these properties of monotonicity in great detail.

Paper Structure

This paper contains 13 sections, 14 theorems, 68 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional Brownian motion with Hurst index H from [0,1]; some properties
  4. Wiener integrals w.r.t. fBm and their square characteristics
  5. Continuity of the variance of EWIFG-process at H=1/2
  6. Extension of EWIFG-process and its square characteristics to the cases H=1 and H=0
  7. Wiener integral w.r.t. fBm with H=1
  8. Wiener integral w.r.t. fBm with $H=0$
  9. Entropy of the EWIFG-process as the function of H and t
  10. Entropy of the EWIFG-process as the function of H from [1/2,1]
  11. Entropy of the EWIFG-process as the function of H from [0,1/2]
  12. Monotonicity and asymptotic behavior of variance of EWIFG-process in time
  13. EWIFG-process with exponent Exp(kt)

Key Result

Lemma 2.3

1) For $0<H<1$ and $s, t\ge 0$ the covariance function $\mathsf{R}_{X^H}(t, s)$ of the process $X^H$ can be presented as In particular, the variance of $X_t^H,\ t\ge 0$, can be calculated as 2) For $1/2<H<1$ and $s, t\ge 0$ the covariance function $\mathsf{R}_{X^H}(t,s)$ of the process $X^H$ can be presented as In particular, the variance of $X_t,\ t\ge 0$, can be calculated as

Figures (5)

  • Figure 1: Graphs of $\frac{\partial \mathsf{V}(1,H)}{\partial H}$ and $\frac{\partial \mathsf{V}(1/2,H)}{\partial H}$
  • Figure 2: Graphs of $\mathsf{V}(H,t)$ as functions of $H\in(\frac{1}{2},1)$
  • Figure 3: Graphs of $\mathsf{V}(H,t)$ as functions of $H\in(0,\frac{1}{2})$
  • Figure 4: Graphs of $\mathsf{V}(H,t)$ as functions of $H\in(0,1)$
  • Figure 5: Surface plot of $\mathsf{V}(H,t)$ as a function of $(H,t)$

Theorems & Definitions (35)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Theorem 2.7
  • ...and 25 more