BoI. Soc. Esp. Ma . ApI.
n051(2010), 71-79
ON
FRACTIONAL
BROWNIAN
MOTIONS
AND
RANDOM
DYNAMICAL
SYSTEMS
MARIA
J.
GARRIDO-ATIENZA*
AND
BJORN
SCHMALFUg
*Dp o.
EDAN,
Uni e si y
o
Se illa
Ap do.
1160 41080 Se illa,
Spain
lns i u
u
Ma hema ik
Fakul a
ElM,
Uni e si a
Pade bo n,
Va bu ge
S asse
100, 33098,
Pade bo n,
Ge many
[email p o ec ed]
[email p o ec ed]
Abs ac
In
his
pape
we conside a class
o
nonlinea
s ochas ic
pa ial
di e en ial
equa ions
(SPDEs)
d i en
by
a ac ional
B ownian
mo ion
wi h
he
Hu s
pa ame e
bigge
han
1/2.
We
show
ha
hese
SPDEs
gene a e
andom
dynamical
sys ems.
Key
wo ds:
F ac ional B ownian mo ions, Random dynamical sys ems,
S ochas ic di e en ial equa ions.
AMS
subjec
classi ica ions:
60H15,
37Hl0,
60H05.
1
In oduc ion
A cen al
ma hema ical
objec in S ochas ics
and
S ochas ic P ocesses
is
he
I o
in eg al.
I
plays
an
impo an
ole in
many
a eas o pu e
and
applied
ma hema ics
including
ma hema ical
inance,
popula ion
dynamics,
luid dynamics, s a is ics, signal p ocessing, con ol, pa icle sys ems,
o
name
a ew.
The
in eg a o
o such
an
in eg al
is
o en chosen
o
be
he
B ownian
mo ion ( he Wiene p ocess) o
i s
semima ingale gene aliza ions. These
andom
unc ions a e o
unbounded
o al
a ia ion, so
ha
hei
S iel jes
in eg als do
no
exis . Special p ope ies o
he
in eg a o s
and
he
in eg ands
a e necessa y
o
gene alize
he
de ini ion o
he
S iel jes in eg al
o
he
I o
in eg al,
and
enable
he
de ini ion o solu ions o di e en ial equa ions d i en
by B ownian mo ion.
A
p ope y
o
pa amoun
impo ance
o
his
e ec o B ownian mo ion
is
he
independence o i s inc emen s. To mo e beyond in eg als
and
p ocesses
cons uc ed
using
his
p ope y
is one o
he
mos
impo an
asks
in
he
heo y
o S ochas ics. We a e mos in e es ed in using
he
ac ional B ownian
mo ion ( E n) p ocess
BH
whe e H E (0,1) is ixed.
I
is a
ype
o s ochas ic
p ocess which de ia es signi ican ly om B ownian mo ion
and
semima ingales.
71
72 M.J. Ga ido-A ienza,
B.
Schmal uR
As a cen e ed Gaussian p ocess, i
is
cha ac e ized by
he
s a iona i y
o i s
inc emen s
and
a medium- o long-memo y
p ope y
which
is
in
sha p
con as
wi h ma ingales
and
Ma ko p ocesses.
I
also exhibi s powe scaling
and
pa h
egula i y p ope ies
wi h
Holde
pa ame e
H,
which
a e
e y dis inc om
B ownian mo ion (no e
ha
he
B ownian mo ion is included in
his
amily o
models when conside ing H =
1/2).
F ac ional B ownian mo ion has become
a
popula
choice o
la e
o applica ions whe e classical p ocesses
canno
model
hese non- i ial p ope ies; o ins ance long memo y, which
is
also known
as pe sis ence,
and
co esponds
o
he
case H E
(1/2,1),
is o undamen al
impo ance
o inancial
da a
and
in
in e ne
a ic, see
[12],
[16]
. F ac ional
B ownian mo ion
is
also a good
candida e
o
model
andom
long
ime
in luences
in clima e sys ems, see
[15].
E e since
he
pionee ing wo ks o Ziihle
[17],
Dec euse ond
and
Us iinel
[5],
and
Lyons
[11],
he
main
h us
has been
o
unde s and
how
o
pe o m
s ochas ic in eg a ion
wi h
espec
o
Bm in a way which
is
consis en
wi h
some p ope ies o
he
classical
I o
heo y
o B ownian mo ion.
In
he
case o
highe egula i y
(H
>
1/2),
simple
ajec o ial
me hods, labelled as pa hwise,
can
be used which make
i
easy
o
ansla e
one in eg a ion
heo y
in o
ano he ,
as ac ional de i a i es allow a pa hwise
es ima e
o
he
in eg als in
e ms
o
in eg and
and
in eg a o
using special no ms. Pa hwise in eg als his o ically
ga e
he
i s cases whe e
adequa e
solu ions
o
s ochas ic di e en ial equa ions
(SDEs) we e es ablished, e.g.
Nuala
and
Rascanu
[14];
in ini e-dimensional
equa ions ha e been
ea ed
wi h
he
same success as ini e-dimensional ones,
e.g.
Nuala
and
Maslowski
[13],
Ga ido-A ienza
e
al.
[6].
In
his
pape ,
we
aim
o
in es iga e
he
equa ions' asymp o ics.
The e
a e
wo heo ies dealing wi h
he
asymp o ic
quali a i e beha io o gene al SDEs:
he
heo y
o
andom
dynamical sys ems (RDS)
and
he
heo y
o exis ence
and
uniqueness o in a ian measu es o
he
associa ed Ma ko semig oup.
Howe e , simila ly
o
Bm i sel , equa ions d i en by Bm do
no
gene a e a
Ma ko p ocess;
his
p ecludes
he
s udy
o in a ian measu es using classical
ools o Bm-d i en sys ems.
This
mo i a es
ou
plan
o
concen a e on
he
s udy
o Bm-d i en SDEs as RDS.
The
heo y
o RDS, de eloped by
L.
A nold
and
cowo ke s, see
[1],
can
be
used
o
desc ibe
he
asymp o ical
and
quali a i e beha io o sys ems o
andom
and
s ochas ic di e en ial/di e ence
equa ion
in
e ms
o s abili y, Lyapuno
exponen s, in a ian mani olds,
and
a ac o s.
As
we
ha e said, conside ing Bm ins ead o B ownian mo ion has some
ad an ages because o
he
nice p ope ies
ha
he
Bm enjoys
and
he
B ownian
mo ion does no .
Ano he
c ucial ad an age
is
he
ollowing: o
many
B ownian-d i en
SPDEs
wi h
non- i ial
di usion coe icien s,
i
is
no
known i
hese equa ions gene a e a RDS.
The
eason is
ha
usually s ochas ic di e en ial
equa ions a e only de ined almos su ely whe e
he
excep ional se may
depend
on w since
his
excep ional se is ela ed
o
he
de ini ion o
an
I o
in eg al which
is
de ined as a limi o
andom
a iables in p obabili y.
And
such a amily o
excep ional se s does
no
allow
o
use
he
heo y
o RDS.
Bu
we
can
o e come
such excep ional se s dealing
wi h
SPDEs
d i en by a Bm
wi h
H >
1/2,
On
F ac ional B ownian mo ions and andom dynamical sys ems
73
p o ided
he
s ochas ic in eg als
a e
in e p e ed
in
he
pa hwise sense.
2
P elimina ies
on
andom
dynamical
sys ems
In
his
sec ion
we
e iew some basic concep s
and
esul s on
andom
dynamical
sys ems
ha
will be used la e .
In
he
nex
de ini ion,
we
in oduce a sys em
ha
models
he
e olu ion o a
noise.
De ini ion
1 A
me ic
dynamical
sys em
(0,
F,
lP',
{e
hE1 ) wi h wo-sided
ime
'll'
(which is
JR.
in
he con inuous case
and
Z
in
he disc e e one) consis s
o
a
p obabili y space (0,
F,
lP')
and
a amily
o
ans o ma ions
{ed E1 such ha :
1.
I
is a one-pa ame e g oup, i.e.
2.
( , w) E
'll'
x 0
---7
e w is measu able,
3.
lP'
is
in a ian
wi h espec o e, i.e.,
e lP'
=
lP',
o all E
'll',
which
means
ha lP'(e A) = lP'(A),
o
all A E F
and
all E
'll'.
4.
lP'
is e godic wi h espec o e, i.e,
o
any
{e hE1 -in a ian
se
BE
F,
which
means
ha
e B
= B
o
all E
'll',
we ha e ei he
lP'(B)
= 0
o
lP'(B)
= 1.
We now in oduce a couple o examples o me ic dynamical sys ems.
Le
V =
(V,
11·11,
e,
.))
be a sepa able Hilbe space.
Conside i s
he
B ownian mo ion. We choose o 0
he
se o con inuous
unc ions Cd' =
Co
(JR.,
V)
on
JR.
wi h alues in V which
a e
ze o
a
ze o.
On
his
se
we
in oduce
he
compac
open
opology gi en by
he
uni o m con e gence
on
compac
in e als in R
The
Bo el-o--algeb a o e
his
space
is
deno ed by
B (Cd').
lP'l
is
he
Wiene measu e.
The
exis ence o such a canonical p ocess
2
(Cd', B(
Cn,
lP'l) ollows by Kolmogo o 's
heo em
abou
he
exis ence o a
2
con inuous modi ica ion o a p ocess, see
Baue
[2].
The
low e
is
gi en by
e we)
=
we
+
)
-
w( ),
wEO
(1)
which is called
he
Wiene shi .
The
Wiene shi
is
measu able, see A nold
[1]
Page
544, because Cd' is sepa able
and
( , w) -7 e w is con inuous. We
emphasize
ha
his
me ic dynamical sys em is e godic, see Boxle
[3].
Now le us in oduce
he
ac ional B ownian mo ion. Gi en H E
(0,1),
a
con inuous cen e ed Gaussian p ocess
3H
( ), E
JR.,
wi h
he
co a iance unc ion
, s E
JR.
is
called a
wo-sided
one-dimensional
ac ional
B ownian
mo ion
( E n),
and
H is
he
Hu s
pa ame e .
Assume
ha
Q is a
bounded
and
symme ic linea
ope a o
on V which
is
o
ace
class, i.e.,
he e
exis a comple e
o hono mal
basis {ediEN in V
and
a
74
M.J. Ga ido-A ienza, B. Schmal uR
sequence o nonnega i e numbe s {AdiEN such
ha
Q
=
L:~l
Ai
<
(X)
and
Qei = Aiei, i E
N.
A con inuous
V- alued
ac ional
B ownian
mo ion
BH
wi h
inc emen al co a iance
ope a o
Q
and
Hu s
pa ame e
H is de ined by
co
BH
( ) = L
V>:;ei/3
( ),
i=l
whe e
{;3 ( )}iEN
is
a sequence o s ochas ically independen one-dimensional
Bm. No ice
ha
he
abo e se ies is con e gen in
L2(0,
F,
JID)
since
L:~l
Ai
<
(X)
and
1E(;3 ( ))2 =
I l
2H
o E R
Rema k
1
Bl/2
is he
B ownian
mo ion.
Using
he
de ini ion o BH, Kolmogo o 's
heo em
ensu es
ha
BH
has a
con inuous e sion.
Thus
we
can
conside
he
canonical
in e p e a ion
o
an
Bm: le 0 = Co(lR,
V),
equipped again
wi h
he
compac
open opology.
Le
F be
he
associa ed Bo el-o--algeb a
and
JID
H
he
dis ibu ion
o
he
Bm BH,
and
{e hER be
he
low o Wiene shi s de ined by (1).
Then
he
quad uple
(0,
F,
JID,
e)
is a me ic dynamical sys em which
is
e godic, see
[9].
Fu he mo e,
We now in oduce
he
concep o
andom
dynamical sys ems
ha
is used
o
desc ibe
he
dynamics o sys ems unde
he
in luence o a noise.
De ini ion
2 A
andom
dynamical
sys em
(RDS) wi h one-sided
ime
']['+
and
phase space V is a
pai
consis ing
o
he
me ic
dynamical
sys em
(0,
F,
JID,
e)
and
a
mapping
cp
:
']['+
x 0 x V
---7
V which is (3(']['+) ® F ®
3(V),
3(V))-
measu able
and
sa is ies he cocycle p ope y
cp
(
,
e T
W,
.)
0
cp
(
T,
W,
.)
=
cp
( +
T,
W,
.),
o
,
T E
']['+,
W
EO,
cp(O,w,·)
= id .
A ypical example o co cycle
mapping
is
he
solu ion
ope a o
o ini e o
in ini e dimensional di e en ial equa ions
wi h
andom
coe icien s sa is ying
pa icula
egula i y assump ions.
Ano he
example
is
he
solu ion
ope a o
o ini e dimensional I o-equa ions. As
we
announced in
he
In oduc ion,
o in ini e dimensional I o-equa ions wi h non- i ial di usion coe icien s
his
p oblem is
a he
unsol ed.
No ice
ha
he
cocycle
p ope y
is
he
gene aliza ion o
he
semig oup
p ope y; in ac , i
we
dele ed all w-dependence in
he
co cycle
p ope y
we
would
jus
ge
he
semig oup p ope y.
We wan
o
s ess
ha
we
ha e equi ed
he
MDS
o
be de ined on wo-
sided
ime
'][',
while
he
RDS
is
only equi ed
o
be de ined on one-sided
ime
']['+.
The
eason is
ha
we
canno
expec
he
mapping
cp
o
be de ined on
'][',
since
i
is gi en, o ins ance, by
he
solu ion
ope a o
o a
SPDE,
which
is
no
in e ible in gene al. Howe e ,
we
can
conside exp essions o
he
ollowing
On
F ac ional B ownian mo ions
and
andom dynamical sys ems
75
ype:
cp( ,e_ w,x), o x E
V,
w E
0,
E
11'+,
exp essions
ha
playa
c ucial
ole
when
analyzing
he
exis ence
o
andom
ixed
poin s
o
andom
a ac o s
associa ed
o
he
RDS
cp,
see
[8].
As we ha e
men ioned,
he
pu pose
o
his
pape
is
o
show
ha
an
in ini e
dimensional
s ochas ic
di e en ial
equa ion
d i en
by
an
Bm
wi h
gene al
di usion coe icien s
gene a es
a
andom
dynamical
sys em.
3
Main
esul s
In
his
sec ion
we i s
in oduce
some basic
concep s
and
esul s
on
ac ional
calculus
and
s ochas ic
in eg als
wi h
espec
o
he
Bm
3H
and
BH.
Fo T > 0,
le
Wa,l(O,
T;
V)
be
he
space
o
measu able
unc ions
:
[0,
T]
---7
V
such
ha
I I =
iT
(1I (s)11
+
is
Il (s)
-
(()11
dl")
ds
<
(X)
a a
(1")00+1"
,
o S 0
s-"
whe e
1-
H < a <
~
is ixed, so we
need
o
conside om now
on
H E
(1/2,1).
Following Ziihle [17], o E
wa,l
(0,
T;
V)
we de ine
he
s ochas ic
in eg al
as
he
gene alized S iel jes
in eg al
!aT
d 3H
=
(-l)a!aT
Do+ (s)D~-=-a 3! -(s)ds,
(3)
i
d 3H
= !aT 1(s, )d 3H, o 0::; s <
::;
T,
whe e, in gene al, o °
::;
a < b
::;
T,
3 !-
(s)
:=
3H
(s)
-
3H
(b),
and
o a < < b
he
Weyl de i a i es
a e
gi en
by
a 1
( ( )
j
( )
-
(()
)
Da+ ( ) =
(l
_
a)
(
_ a)a + a a
(
_
()a+1
d(
,
D1- a
3H
( ) = (_1)1-00 ( 3H( ) - 3H(b)
+(l-a)l
b
3H( )
- 3
H
(()d()
b- b-
(a)
(b
-
)1-a
((
-
)2-a
'
whe e
deno es
he
Gamma
unc ion.
I
can
be
p o ed
(see, o
ins ance,
Nuala
and
Ra§canu
[14], Dec euse ond
and
Us iinel
[5],
Ziihle [17])
ha
he
s ochas ic
in eg al
(3) exis s.
Now we de ine
he
s ochas ic
in eg al
wi h
espec
o
he
in ini e
dimensional
Bm BH.
Le
L(V)
deno e
he
space
o
linea
bounded
ope a o s
on
V
and
le
G : 0 x
[0,
T]
---7
L(V)
be
an
ope a o
such
ha
G(w, ·)ei E
wa,l(o,
T;
V) o
each
i E
Nand
w E
o.
We de ine
whe e
he
con e gence
o
he
sums
in (4) is
unde s ood
in
V.

76 M.J. Ga ido-A ienza,
B.
Schmal uR
The
ollowing esul es ablish
ha
when making a change o a iable in
he
s ochas ic in eg al,
we
no
only ha e
o
shi
he
in eg a ion in e al
and
he
a iable
bu
also
he
pa h
o
he
Bm ( o
he
p oo , see
[6]).
Lemma
1 Fo
a,
b,
E
JR.,
assuming ha
bo h
in eg als
a e
well-de ined,
Ib
Ib-
a G(s)dw(s) = a- G(s + )de w(s).
Conside now
he
ollowing s ochas ic e olu ion
equa ion
in V
{ du( ) = (Au( ) + F(u( )))d + G(u( ))dw( ),
u(O)
=
Uo
E V
whe e w deno es
he
in ini e dimensional Bm
BH
(see (2)).
(5)
Assume
ha
A
is
he
in ini esimal
gene a o
o
an
analy ic semig oup
Se),
and
ha
F : V --7 V is Lipschi z con inuous wi h Lipschi z
cons an
LF,
and
G : V --7
L(V)
and
G'
: V --7 L(V,
L(V))
a e
Lipschi z con inuous in
he
ollowing senses:
sup
IIG( dei -G( 2)eill
::;
Lcll l
-
211,
iEN
sup
IIG'( dei -G'( 2)eiIIL(V)
::;
L~ll l
-
211,
iEN
(6)
(7)
whe e {ediEN is
he
comple e
o hono mal
basis in V in oduced in Sec ion
2.
The
solu ion o (5) on
[0,
T] is a V- alued p ocess u whose
pa hs
a e o
e e y
wE
0 elemen s o
Woo,l(O,
T;
V),
o
an
a E
(1
-H,
~),
and
u( ) =
S( )uo+
!a
S( -s)F(u(s))ds+
!a
S( -s)G(u(s))dw,
E
[O,T],
(8)
whe e
he
s ochas ic in eg al has
o
be
unde s ood
acco ding
o
(4).
Fo such
an
a E
(1
-
H,
~
), deno e by
w ~(X)
(0,
T;
V)
he
Banach
space o
measu able unc ions x :
[0,
T]
--7 V such
ha
(I
(
~
i
Ilx( ) -
x( )11
)
IlxIIOO,~,(I
=
sup
e-
Ilx( )11
+ (
)1+
00
d
<
00
E[O,T] ° -
o
(J
?:
1,
and
~
E
[a,
1 - a).
The
ole o
he
ac o
~
is
c ucial when p o ing
he
ollowing exis ence heo em, which p oo
can
be ound in
[6].
Theo em
2 Le a E
(1
-H,
~),
(J
?:
1 and
~
E
[a,l
-
a).
Assume F is
Lipschi z con inuous, and ha G and
G'
sa is y (6) and (7). Then, o
each
ini ial poin
Uo
E V he e exis s a unique solu ion
o
equa ion (8) wi h i s pa hs
in
w ~(X)(O,
T;
V).
In addi ion, he mapping
<I>
: V --7
w ~(X)(O,
T; V) gi en
by
<I>
:
Uo
-7 u
is
con inuous o w E
o.
On
F ac ional B ownian mo ions
and
andom dynamical sys ems 77
Theo em
3
The
solu ion u o (8) de ines a andom dynamical sys em
cp
:
lR+
x [2 X V
---7
V,
gi en
by
cp( ,
w,
uo)
= S( )uo + a
S(
-s)F(u(s))ds + a
S(
-s)G(u(s))dw.
P oo .
The
measu abili y ollows by
[4]
Lemma
111.14.
T i ially
cp(O,
w,
x)
=
Uo.
Le
us check
hen
he
cocycle p ope y: o
, T E
lR+,
wE
[2
and
Uo
E
V,
we
ha e
+
T
cp( +T,W,UO)=S( +T)UO+
Jo
S( +T-s)F(u(s))ds
+
T
+
Jo
S( +T-s)G(u(s))dw(s)
= S( ) (S(T)UO
+
aT
S(T -s)F(u(s))ds +
aT
S(T -s)G(U(S))dW(S))
I
+T
I +T
+ T
S(
+ T -s)F(u(s))ds + T
S(
+ T -s)G(u(s))dw(s).
Making
he
change o a iable s -T = , applying
Lemma
1,
I
+T
T
S(
+ T -s)G(u(s))dw(s) =
Jo
S(
- )G(u( + T))deTw( ),
and
hen,
se ing
y(s) = u(s +
T),
o s E
[0,
l,
cp(
+
T,
w,
uo)
= S( )y(O) + a
S(
- )F(y( ))d + a
S(
- )G(y( ))deTw( )
=
cp( ,
eTw,') 0
cp(T,
W,
uo).
D
P o ing
ha
ou
s ochas ic
equa ion
(8) gene a es a RDS
is
he
s a ing
poin
o
analyze
i s
asymp o ic
beha io . One possibili y, which is a key concep
desc ibing
he
dynamics o RDS gene a ed by Bm-d i en SDEs,
is
he
so-called
global
a ac o ,
which is
an
in a ian
compac
andom
se
a ac ing
o he
bounded
andom
se s.
The
essen ial dynamics
ake
place in a neighbo hood o
he
a ac o
(see [8]).
Ano he
op ion
o
discuss
he
s abili y o Bm-d i en
SDEs
is
o
s udy
he
exis ence o s able
and
uns able mani olds
and
Lyapuno
exponen s, see
[10]
and
[7].
Such
smoo h
mani olds
a e
in a ian
unde
he
dynamics o
he
sys ems,
and
on
hem,
he
s a es
a e
a ac ed
o epelled by
a
s eady
s a e.
Re e ences
[1]
L.
A nold, Random Dynamical Sys ems, Sp inge Monog aphs in
Ma hema ics,
Sp inge -Ve lag, Be lin 1998.
78
M.J. Ga ido-A ienza,
B.
Schmal uR
[2]
H.
Baue , P obabili y Theo y, de
G uy e
S udies in Ma hema ics. Wal e
de
G uy e
& Co., Be lin 1996.
[3]
P. Boxle , S ochas ische Zen umsmannig al igkei en. Ph.D. hesis,
Ins i u
U
Dynamische Sys eme, Uni e si ii B emen, 1988.
[4]
C. Cas aing,
M.
Valadie , Con ex analysis and measu able mul i unc ions.
Lec u e No es in Ma hema ics,
Vol.
580. Sp inge -Ve lag, Be lin 1977.
[5]
L.
Dec euse ond, A.S. Us iinel, S ochas ic analysis
o
he ac ional
B ownian mo ion. Po en ial Analysis, 10 (1998), p. 177-214.
[6]
M.J. Ga ido-A ienza, K. Lu, B. Schmal uB,
Random
dynamical sys ems
o s ochas ic pa ial di e en ial equa ions d i en
by
a ac ional B ownian
mo ion,
submi ed
(2009).
[7]
M.J. Ga ido-A ienza, K. Lu, B. Schmal uB, Uns able in a ian mani olds
o s ochas ic
PDEs
d i en
by
a ac ional B ownian mo ion. To
appea
in
J.
Di e en ial Equa ions.
[8]
M.J. Ga ido-A ienza, B. Maslowski, B Schmal uB,
Random
a ac o s o
o dina y s ochas ic equa ions d i en
by
a ac ional B ownian
mo ion
wi h
Hu s pa ame e g ea e han
1/2.
To
appea
in In e na ional
Jou nal
on
Bi u ca ion
and
Chaos.
[9]
M.J. Ga ido-A ienza, B. Schmal uB, E godici y
o
he in ini e dimensional
ac ional B ownian mo ion,
submi ed
(2009).
[10]
K. Lu, B. Schmal uB, In a ian mani olds o s ochas ic wa e equa ions. J.
Di e en ial Equa ions, 236 (2007),
2,
p. 460-492.
[11]
T.
Lyons, Di e en ial equa ions d i en
by
ough signals. Re . Ma .
Ibe oam., 14 (1998),
2,
p. 215-310.
[12]
B.B. Mandelb o ,
J.W.
an Ness. F ac ional B ownian mo ions, ac ional
noises and applica ions. SIAM Re iew, 10 (1968), p. 422-437.
[13]
B. Maslowski, D. Nuala , E olu ion equa ions d i en
by
a ac ional
B ownian mo ion.
Jou nal
o Func ional Analysis, 202 (2003), p. 277-305.
[14]
D. Nuala ,
A.
Rascanu, Di e en ial equa ions d i en
by
ac ional
B ownian mo ion. Collec anea Ma hema ica,
53
(2002), p. 55-8l.
[15]
T.
N.
Palme , G.
J.
Shu s, R. Hagedo n, F. J. DobIas-Reyes,
T.
Jung,
M.
Leu beche , Rep esen ing model unce ain y in wea he and clima e
p edic ion. Annu. Re .
Ea h
Plane . Sci., 33 (2005), p. 163-193.
[16]
W. Willinge ,
M.
Taqqu,
V.
Te e o sky, Long ange dependence and s ock
e u ns. Finance
and
S ochas ics, 3 (1999), p. 1-13.
[17]
M.
Ziihle, On he link be ween ac ional and s ochas ic calculus. S ochas ic
dynamics (B emen, 1997), p. 305-325, Sp inge , 1999.