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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�

(2)

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

(3)

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.

(4)

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.

(5)

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.

(6)

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

(7)

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

(8)

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

(9)

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

(10)

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)

(11)

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.

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