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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=∞

0uH−(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.