HAL Id: hal-00539236
https://hal.archives-ouvertes.fr/hal-00539236
Submitted on 24 Jan 2011
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires
From N parameter fractional Brownian motions to N parameter multifractional Brownian motions
Erick Herbin
To cite this version:
Erick Herbin. From N parameter fractional Brownian motions to N parameter multifractional Brow- nian motions. Rocky Mountain Journal of Mathematics, Rocky Mountain Mathematics Consortium, 2006, 36 (4), pp.1249-1284. �hal-00539236�
to N parameter multifrational Brownian
motions
Erik Herbin
INRIA,DomainedeVolueau,Roquenourt, BP105,78153 LeChesnayCedex,Frane
erik.herbininria.fr
and
DassaultAviation,78quaiMarelDassault,92552Saint-CloudCedex,Frane
erik.herbindassault-aviation.fr
July 11,2002
Abstrat
Multifrational Brownian motion is an extension of the well-known
frationalBrownianmotionwheretheHolderregularityisallowedtovary
alongthe paths. Inthis paper, two kindof multi-parameterextensions
ofmBmarestudied: oneisisotropi while theotherisnot. For eahof
theseproesses, amovingaverage representation, aharmonizablerepre-
sentation,andtheovarianestruturearegiven.
TheHolder regularity is then studied. Inpartiular, the ase of an ir-
regularexponentfuntionHisinvestigated. Inthissituation,thealmost
surepointwiseand loalHolderexponentsofthemulti-parametermBm
areprovedtobeequaltotheorrespondentexponentsofH. Eventually,
aloal asymptoti self-similarity property is proved. The limit proess
anbeanotherproessthanfBm.
AMSlassiation: 62G05,60G15,60G17,60G18
Key words: frational Brownian motion, Gaussian proesses, Holder regu-
larity,loalasymptotiself-similarity,multi-parameterproesses
1 Introdution
Inmanyappliations,frationalBrownianmotion(fBm)seemstotverywell
torandomphenomena. Reallthatitanbedenedbyoneofthefourfollowing
properties. LetH 2(0;1)(H issometimesalledtheHurstparameter).
B H
isaenteredGaussianproesssuhthat
8s;t2R
+
; E
B H
s B
H
t
= 1
2
s 2H
+t 2H
jt sj 2H
theproessB suh that
8t2R
+
; B H
t
= Z
0
1 h
(t u) H
1
2
( u) H
1
2 i
:W(du)+
Z
t
0 (t u)
H 1
2
:W(du)
isafBm,
theproessB H
suh that
8t2R
+
; B H
t
= Z
R e
it
1
jj H+
1
2 :
^
W(d)
isafBm,
B H
istheuniqueself-similarGaussianproesswithstationaryinrements.
ItseÆieny hasalreadybeenshownin simulation oftraÆ onInternetor
in nane. This indued some reent progress suh as stohasti integration
againstfBm.
However, themain limitation of fBmis that theHolderregularity isonstant
alongthepaths.
MultifrationalBrownianmotion(mBm)hasbeenindependentlyintroduedin
[4℄ and [13℄. This proess is a generalization of frational Brownian motion
where the Hurst parameter H is substituted by a funtion t 7! H(t). As a
onsequenetheHolderexponentisallowedto varyalongtrajetories.
The dierent denitions by the two groupsof authors provided two dierent
representationsofmBm.
Peltier and Levy-Vehel ([13℄) dened the mBm from the moving average
denition ofthefrationalBrownianmotion
X
t
= Z
0
1 h
(t u) H(t)
1
2
( u) H(t)
1
2 i
:W(du)+ Z
t
0 (t u)
H(t) 1
2
:W( du)
wheret7!H(t)isaHolderfuntion.
Benassi, Jaard and Roux ([4℄) dened the mBm from the harmonizable
representationofthefBm
X
t
= Z
R e
it
1
jj H(t)+
1
2 :
^
W(d)
These two denitions were proved to be equivalent up to amultipliative
deterministifuntion ([6℄).
Moreover,in [3℄ the ovarianefuntion of this Gaussian proess hasbeen
provedtobe
E[X
s X
t
℄=D(H(s);H(t)) h
jsj
H(s)+H(t)
+jtj
H(s)+H(t)
jt sj
H(s)+H(t) i
whereD isaknowndeterministifuntion.
Thegoalofthis paperisto study somemulti-parameter extensionofthemul-
tifrationalBrownianmotion, ieastohastiproess indexed by R N
+
, whih is
anmBmwhenN =1. Oneextensionhasalreadybeenonsideredin[4℄.
2D extension of frational Brownian motionhas beenalready used in various
appliations suh asunderwaterterrain modeling ([14℄). It may bemorereal-
istito allowloal regularity tovary at eah point : ourextension ofmBm in
R 2
maybeused forthis kindofappliation.
Brownian motion
Sine multifrational Brownian motion is anextension of frational Brownian
motion, westart withareview oftheexisting extensions offBm. Most ofthe
resultsinthissetionarewell-known,butwegivenewproofsbasedonlyonthe
ovarianefuntions.
In the same way as Brownian motion has two main multi-parameter ex-
tensions: Levy Brownian motion and Brownian sheet, two dierent multi-
parameterextensionsoffrationalBrownianmotionhavebeendened.
2.1 Levy frational Brownian motion
The Levy frational Brownian motion is dened to be a entered Gaussian
proessofovarianefuntion
E[X
s X
t
℄= 1
2
ksk 2H
+ktk 2H
kt sk 2H
(1)
Thereareseveraldenitionsofthisproessbyitstrajetories. Amongthese,
itanbedenedasintegralagainstwhitenoise. Lindstromstatedthefollowing
(see[9℄).
Proposition1 The proessdenedby
X
t
= Z
R N
h
kt uk H
N
2
kuk H
N
2 i
W(d u) (2)
isaLevyfrational Brownian motion uptoamultipliative onstant.
TheharmonizablerepresentationoffrationalBrownianmotionanalsobe
generalized.
Proposition2 The proessdenedby
X
t
= Z
R N
e iht;i
1
kk H+
N
2 :
^
W(d) (3)
where
^
W isthe Fouriertransformof whitenoise inR N
,
isaLevyfrational Brownian motion uptoamultipliative onstant.
Proof As will bedone formultifrational Brownian eld, the Fouriertrans-
formofthekernelofrepresentation(2)ouldbediretlyomputed. Butasthis
representation denes a real entered Gaussian proess, it is enough to show
that theovarianefuntionhastheform(1).
Forallt 2R N
,let'sdenote byf
t
thefuntion 7!
e i<t; >
1
kk H+
N
2
andonsider the
entered GaussianproessX = n
X
t
=
^
W(f
t );t2R
N
+ o
.
First,weremarkeasilythat forallt,almostsurely,
^
W(f
t )2R.
E[X
s X
t
℄ = E h
^
W(f
s )
^
W(f
t )
i
= Z
R N
e i<s;>
1
e i<t;>
1
kk 2H+N
:d
= Z
R N
e
i<s t;>
e i<s;>
e i<t;>
+1
kk 2H+N
:d
Thenwehavetoonsider3integralsoftheform R
R N
1 e i<t; >
kk 2H+N
:d.
Fort2R N
xed,onsiderthehangeofvariablesfromR N
intoitself,u=( )
where is thelinear appliation whih maps the anoni basis of R N
to the
orthonormalbasis
e
1
= t
ktk
;e
2
;:::;e
N
.
Then,weget
Z
R N
1 e i<t;>
kk 2H+N
:d= Z
R N
1 e iktk:u1
kuk 2H+N
:du
Aftertheseondhangeofvariables
v=ktk:u=ktkId:u
dv=ktk N
:du
weget
Z
R N
1 e i<t;>
kk 2H+N
:d= ktk
2H+N
ktk N
Z
R N
1 e iv1
kvk 2H+N
:dv
| {z }
CN;H>0
Proeedingthesamewayforthe2otherintegrals,weanonlude
E[X
s X
t
℄=C
N;H
ksk 2H
+ktk 2H
kt sk 2H
whih shows that the proess
1
p
C
N;H
^
W(f
t );t2R
N
+
is a Levy frational
Brownianmotion. 2
2.2 Frational Brownian sheet
On the ontrary to the Levy frational Brownian motion, this proess is not
isotropi. Inpartiular, weanhavedierentHurst parametersin eah ofthe
N diretions.
ThefrationalBrowniansheet(fBs)isdenedtobeaenteredGaussianproess
ofovarianefuntion
E[X
s X
t
℄= N
Y
i=1 1
2
s 2H
i
i +t
2H
i
i jt
i s
i j
2Hi
(4)
Asintheisotropiase,thisproesshastwodierentrepresentationsbyits
trajetories.
X
t
= Z
R N
N
Y
i=1 h
jt
i u
i j
H
i 1
2
ju
i j
H
i 1
2 i
W( du)
isafrational Brownian sheet, uptoamultipliative onstant.
Remark 1 In[8 ℄, Pontier/Leger introduedanother moving averagerepresen-
tation offrational Brownian sheet.
X
t
= Z
R N
N
Y
i=1 h
(t
i u
i )
Hi 1
2
+
( u
i )
Hi 1
2
+ i
W(d u)
Proof Thisproessis obviouslyGaussian and entered. Thus, we onlyneed
toshowthat itsovarianefuntion hastheexpetedform. Weompute
E[X
s X
t
℄= N
Y
i=1 Z
R h
js
i u
i j
Hi 1
2
ju
i j
Hi 1
2 ih
jt
i u
i j
Hi 1
2
ju
i j
Hi 1
2 i
:du
i
Weansee thatthefatororrespondingtoeahi, istheovarianeofafBm
withHurst parameterH
i
(oraLevyfrationalBrownianmotionwithN =1).
Thenwehave
E[X
s X
t
℄= N
Y
i=1 K
1;Hi
js
i j
2H
i
+jt
i j
2H
i
jt
i s
i j
2H
i
2
This proess also has an harmonizable representation, using the Fourier
transformofthewhitenoiseinR N
asinthepreviousparagraph.
Proposition4 Forall t=( t
i
) , onsider the funtion
t
suhthat for all =
(
i ) ,
t (u)=
N
Y
m=1 e
it
m
m
1
j
m j
Hm+
1
2
The proessdenedby
X
t
=
^
W(
t )=
Z
R N
N
Y
m=1 e
itmm
1
j
m j
Hm+
1
2
^
W(d)
isafrational Brownian sheet, uptoamultipliative onstant.
Proof As in the previousproposition, let'sompute theovarianefuntion
ofthisproess.
E[X
s X
t
℄ = N
Y
m=1 Z
R e
ismm
1
e itmm
1
j
m j
2H
m +1
:d
m
= N
Y
m=1 C
1;Hm
js
m j
2Hm
+jt
m j
2Hm
jt
m s
m j
2Hm
usingthesameargumentofthepreviousproposition. 2
the samelaw. Infat, as apartiularaseof proposition10,they areindistin-
guishable.
2.3 Stationarity of inrements and self similarity
Letusstartbyreallingthenotionofinrementsin R N
+ .
Forafuntion f : [0;1℄
N
!R andh 2R, oneusually dene theprogressive
diereneindiretion
i by
h;i f(x)=
f(x+h
i
) f(x) ifx;x+h
i 2[0;1℄
N
0 either
andforh2R N
andA=(i
1
;:::;i
k ),
h;A f =
hi
1
;i1
f ÆÆ
hi
k
;ik f
DespitethetemptationtodenetheinrementsbyX
t X
s
asinonedimension,
itisbettertoset
X
s;t
=
t s;(1;:::;N) X
s
=
X
r2f0;1g N
( 1) N
P
l rl
X
[si+ri(ti si)℄
i
(5)
If thereexists i2f1;:::;Ngsuhthat s
i
=t
i
, wehaveX
s;t
=0. Then, we
onsider
I =fi=1;:::;N; s
i 6=t
i g
and
t s;I X
s
= X
r2f0;1g
#I ( 1)
#I P
l r
l
X
[si+ri(ti si)℄
i2I
2.3.1 Isotropi ase
Intheisotropiase,thefollowingextensionoffBm'spropertiesarewellknown
(see[9℄).
Proposition5 LetX= n
X
t
;t2R N
+ o
beaLevyfrational Brownian motion.
Wehave the twofollowing properties forall h2R N
+
anda>0
X
t+h X
h (d)
= X
t X
0
X
at (d)
= a H
X
t
where (d)
= meansequality ofnite dimensionaldistributions.
Proposition5impliesthestationarityofinrements(5).
Proposition6 The inrementsofLevyfrational Brownianarestationary,ie
for allh2R N
+
X
h;t+h (d)
=X
0;t
Proof Wexh2R
+
andwrite
X
h;t+h
=
X
r2f0;1g N
f0g ( 1)
N P
l rl
X
[h
i +r
i t
i
℄
i X
h
then in the development of E[X
h;s+h X
h;t+h
℄, we only have terms of the
form
E
X
[hi+risi℄
i X
h
X
[hi+iti℄
i X
h
=E
X
[risi℄
i X
[iti℄
i
usingthepreviousproposition. Thereforewehave
E[X
h;s+h X
h;t+h
℄=E[X
0;s X
0;t
℄
2
2.3.2 Non-isotropiase
Inthenon-isotropiase,thepropertiesofself-similarityandstationarityofin-
rementshavebeenstatedbyLeger/Pontier(f[8℄). Here,wegiveanotherproof
basedontheovarianefuntionratherthanthemovingaveragerepresentation.
Proposition7 Let X = n
X
t
;t2R N
+ o
be a frational Brownian sheet. We
have thetwofollowing propertiesfor all h2R N
+
anda>0
X
h;t+h (d)
= X
0;t
X
at (d)
= a
P
i H
i
X
t
Proof WeonsiderN independentfBmX (1)
;:::X (N)
ofHurstparameterH
i ,
andtheproessY = n
Y
t
;t2R N
+ o
suhthatY
t
= Q
N
i=1 X
(i)
t
i
. Weanseeeasily
that X and Y havethesameovarianefuntion. Thesameresultfollowsfor
theinrements n
X
h;t+h
;t2R N
+ o
and n
Y
h;t+h
;t2R N
+ o
. Asaonsequene,
from
Y
h;t+h
=
X
r2f0;1g N
( 1) N
P
l r
l N
Y
i=1 X
(i)
h
i +r
i t
i
= N
Y
i=1 h
X (i)
t
i +h
i X
(i)
h
i i
weget
E[X
h;s+h X
h;t+h
℄ = N
Y
i=1 E
h
X (i)
si+hi X
(i)
hi
X (i)
ti+hi X
(i)
hi i
| {z }
E h
X (i)
s
i X
(i)
t
i i
= E[X
0;s X
0;t
℄
E[X
as X
at
℄=E h
a P
i Hi
X
s a
P
i Hi
X
t i
2
Therefore,weanonludethatbothextensionsoffBmsatisfytheproperties
ofself-similarityandstationarityofinrements.
3 The multifrational Brownian motion's ase
One again, we an onsider two dierent kinds of multi-parameter exten-
sion of mBm : isotropi and anisotropi extension. Note, rst of all, that
mBm alreadyhasamulti-parameterextension. Indeed,theformulationof Be-
nassi/Jaard/Roux in [4℄ was done for t 2 R N
. We will see that it an be
onsideredasanisotropiextension.
3.1 Isotropi extension
Todene anisotropi extensionof themBm, thenaturalwayis to substitute
the onstant H of the moving average representation of the Levy frational
Brownianmotion,with afuntion.
Denition1 Let H : R N
! (0;1) be a measurable funtion. The proess
n
X
t
;t2R N
+ o
suhthat
X
t
= Z
R N
h
kt uk H(t)
N
2
kuk H(t)
N
2 i
W(d u) (6)
isalledmultifrational Brownian eld.
We will show that this proess is the same as the proess dened by Be-
nassi/Jaard/Roux. This result generalizes on the equivalene stated in the
aseN =1in[6℄.
Proposition8 Let H : R N
! (0;1) be a measurable funtion. The proess
denedby
X
t
= Z
R N
e iht;i
1
kk H(t)+
N
2 :
^
W(d) (7)
isindistinguishable,uptoamultipliative deterministifuntion,fromthe pro-
ess denedby (6). Thisformulation isthe harmonizable representation ofthe
multifrational Brownian eld.
Proof Firstofall,letusomputetheFouriertransformofthefuntionk:k
.
hTk:k
;'i = h k:k
;'i^
= Z
R N
ktk
Z
R N
e i<w;t>
'(w):dw
:dt
R N
R N
! R
N
R N
(w;t) 7! (w;=(t))
where is thelinear appliation whih maps the anoni basis of R N
to the
orthonormalbasis
e
1
= w
kwk
;e
2
;:::;e
N
. Weget
hTk:k
;'i = Z
R N
Z
R N
kk
e i1kwk
'(w):dw:d
= Z
R N
Z
R N
kuk
kwk
e iu1
'(w) dw:du
kwk N
usingthehangeofvariables (w;)7!(w;u=kwk). Thenwehave
h Tk:k
;'i= Z
R N
kuk
e iu1
:du
| {z }
Z
R N
1
kwk +N
'(w):dw
Thus,
Tk:k
(w)=
kwk +N
From this result, an elementary omputation gives the Fourier transform of
kt :k
k:k
. Weget
T [ kt :k
k:k
℄(v)=
e i<t;v>
1
kvk +N
Wededuethat8t2R N
, almostsurely,
Z
R N
h
kt uk H(t)
N
2
kuk H(t)
N
2 i
W(du)=
H(t) Z
R N
e iht;i
1
kk H(t)+
N
2 :
^
W(d)
using thefatwesawpreviouslythat theseondintegralisalmost surelyreal.
Therefore,byanargumentofontinuity,theresultfollows. 2
Thisproessisobviouslyaentered Gaussianproess. It isthusofinterest
to study itsovariane funtion. Thefollowing proposition is an extension of
theaseN=1statedin[3℄.
Proposition9 Let n
X
t
;t2R N
+ o
be a multifrational Brownian eld. There
existsa deterministifuntion D f
N
:R!R suhthat the ovariane funtion
of X anbewritten
E[X
s X
t
℄=D f
N
(H(s)+H(t)) h
ksk
H(s)+H(t)
+ktk
H(s)+H(t)
kt sk
H(s)+H(t) i
(8)
sentation. Bydenitionof
^
W,wehave
E[X
s X
t
℄= Z
R N
e i<s;>
1
e i<t;>
1
kk
H(s)+H(t)+N
:d
ThisintegralhasalreadybeenalulatedforaLevyfrationalBrownianmotion
withaparameterH =
H(s)+H(t)
2
. Thenwehave
E[X
s X
t
℄= Z
R N
1 e iu1
kuk
H(s)+H(t)+N :du
| {z }
D f
N
(H(s)+H(t))
h
ksk
H(s)+H(t)
+ktk
H(s)+H(t)
kt sk
H(s)+H(t) i
withD f
N (x)=
R
R N
1 e iu
1
kuk x+N
:du2
3.2 Non isotropi extension
Anotherwaytoextendthemultifrational Brownianmotionforasetofindex
inludedinR N
+
,istoopythedenitionoftheBrowniansheet.
Denition2 Let H : R N
+
! (0;1) N
be a measurable funtion. The proess
n
X
t
;t2R N
+ o
suhthat
X
t
= Z
R N
N
Y
i=1 h
jt
i u
i j
H
i (t)
1
2
ju
i j
H
i (t)
1
2 i
W( du)
whereW isthe whitenoise, isalledmultifrational Brownian sheet(mBs).
As in the ase of the isotropi extension, there also exists a harmonizable
representationofthemBs.
Proposition10 Let H : R N
+
! (0;1) N
be a measurable funtion. For all
t=(t
i )
i2f1;:::;Ng
,weonsider the funtion
t
suhthat forall =(
i ),
t (u)=
N
Y
m=1 e
it
m
m
1
j
m j
Hm(t)+
1
2
The proessdenedby
X
t
=
^
W(
t )=
Z
R N
N
Y
m=1 e
it
m
m
1
j
m j
H
m (t)+
1
2
^
W(d)
isindistinguishable,uptoamultipliative deterministifuntion,fromthe pro-
ess dened previously. Thisformulation isthe harmonizable representation of
the multifrational Brownian sheet.
