Fractionally integrated Bessel process

Abstract

We consider a fractionally integrated Bessel process defined by Y s δ , H = ∫ 0 ∞ ( u H − ( 1 / 2 ) − ( u − s ) + H − ( 1 / 2 ) ) d X u δ , where X δ is the Bessel process of dimension δ > 2. We discuss the relation of this process to the fractional Brownian motion at its maximum, study the basic properties of the process and prove its Hölder continuity.

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royalsocietypublishing.org/journal/rspa
Research
Cite this article: Shevchenko G, Zatula D.
2021 Fractionally integrated Bessel process.
Proc.R.Soc.A477: 20200934.
https://doi.org/10.1098/rspa.2020.0934
Received: 24 November 2020
Accepted: 20 May 2021
Subject Areas:
statistics
Keywords:
fractional Brownian motion (fBm), Hurst
parameter, Bessel process, Hölder continuity
Author for correspondence:
Georgiy Shevchenko
e-mail: zhora@univ.kiev.ua
Fractionally integrated Bessel
process
Georgiy Shevchenko1and Dmytro Zatula2
1Kyiv School of Economics, 3 Shpaka str., 03113 Kyiv, Ukraine
2Department of Computational Mathematics, Taras Shevchenko
National University of Kyiv, 60 Volodymyrska str., Kyiv, Ukraine
GS, 0000-0003-1047-3533;DZ,0000-0002-3371-9287
We consider a fractionally integrated Bessel process
defined by Yδ,H
s=∞
0uH−(1/2) −(u−s)H−(1/2)
+dXδ
u,
where Xδis the Bessel process of dimension δ>
2. We discuss the relation of this process to the
fractional Brownian motion at its maximum, study the
basic properties of the process and prove its Hölder
continuity.
1. Introduction
The starting point of this article is the conjecture
formulated in ref. [1] concerning the behaviour of frac-
tional Brownian motion (fBm) BHin the neighbourhood
of its point of maximum. Loosely speaking, it says that
the path of fBm near its maximum is ‘pointed’. Precisely,
if t∗=arg maxt∈[0,1]BH
t,foranyε>0, there exists δ>0
such that
BH
t≤BH
t∗−|t∗−t|H+ε(1.1)
for all t∈[0, 1] with |t∗−t|≤δ.
A proof of this property is available only for H=1/2:
in this case, fBm BH=Wis a standard Brownian motion,
and it is well known that it behaves like a Bessel process
of dimension 3 in the neighbourhood of its maximum
point, i.e.
Wt∗−Wt∗−εs
ε1/2,s≥0→X3
s,s≥0,ε→0+, (1.2)
where X3is Bes(3) process started at zero. The
proof (see e.g. ref. [2], Theorem 1) is based on
the decomposition theorem for Brownian motion, in
particular, it relies heavily on the Markov structure
of Brownian motion. However, for H=1/2, fBm
is not a Markov process, so there is no way to
generalize the existing techniques in order to get (1.1).
2021 The Author(s) Published by the Royal Society.All rights reser ved.