Continuous-time autoregressive moving-average processes in Hilbert space

0.15 MB·

by  Fred Espen Benth

Continuous-time autoregressive moving-average
processes in Hilbert space
FredEspenBenthandAndre´Su¨ss
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1 AbstractWeintroducetheclassofcontinuous-timeautoregressivemoving-average
(CARMA) processes in Hilbert spaces. As driving noises of these processes we
]
R consider Le´vy processes in Hilbert space. We provide the basic definitions, show
relevantpropertiesoftheseprocessesandestablishtheequivalentsofCARMApro-
P
. cessesontherealline.Finally,CARMAprocessesinHilbertspacearelinkedtothe
h
stochasticwaveequationandfunctionalautoregressiveprocesses.
t
a
m
[
1 Introduction

1
v
8 Continuous-time autoregressive moving-averageprocesses, or CARMA for short,
1 playanimportantroleinmodellingthestochasticdynamicsofvariousphenomena
6
likewindspeed,temperaturevariationsandeconomicindices.Forexample,based
4
on such models, in [1] the author analyses fixed-incomemarkets while in [8] and
0
[13] the dynamics of weather factors at various locations in Europe and Asia are
.
1
modelled. Finally, in [7], [4] and [17] continuous-time autoregressive models for
0
commoditymarketslikepowerandoilarestudied.
7
1 CARMA processes constitute the continuous-time version of autoregressive
: moving-averagetimeseriesmodels.Inthispaperwegeneralizetheseprocessesto
v
a Hilbertspace context.The crucialingredientin the extensionis a ”multivariate”
i
X Ornstein-Uhlenbeck process with values in a Hilbert space. There already exists
r ananalysisofinfinitedimensionalLe´vy-drivenOrnstein-Uhlenbeckprocesses,and
a
we refer the readerto the survey[3]. Moreover,matrix-valuedoperatorsand their
FredEspenBenth
University of Oslo, Department of Mathematics, POBox 1053 Blindern, N-0316 Oslo, Norway
e-mail:[email protected]
AndreSu¨ss
Universitat de Barcelona, Facultat de Matematiques, Gran Via, 585, E-08007 Barcelona, Spain
e-mail:[email protected]
1
2 FredEspenBenthandAndre´Su¨ss
semigroups play an important role. In [12] a detailed semigroup theory for such
operatorsisdeveloped.Wereviewsomeoftheresultsfrom[3]and[12]inthecon-
textofHilbert-valuedCARMAprocesses,aswellasprovidingsomenewresultsfor
theseprocesses.
Letusrecallthedefinitionofareal-valuedCARMAprocess.Wefollow[11]and
first introducethe multivariate Ornstein-Uhlenbeckprocess Z(t) with values
t 0
inRp for p Nby { }≥
∈
dZ(t)=C Z(t)dt+e dL(t), Z(0)=Z Rp. (1)
p p 0
∈
Here,Lisaone-dimensionalsquareintegrableLe´vyprocesswithzeromeandefined
onacompleteprobabilityspace(W ,F,P)withfiltrationF = F ,satisfying
t t 0
the usual hypotheses. Furthermore, e is the ith canonical unit{vec}to≥r in Rp, i=
i
1,...,p.The p pmatrixC takestheparticularform
p
×
0 1 0 . .. 0
0 0 1 0.. 0
 
. . . . .. .
C = , (2)
p  . . . . .. . 
 
 0 . . . .. 1 
 a p a p 1 . . .. a 1
− − − − 
 
forconstantsa >0,i=1,...,p.1
i
Wedefineacontinuous-timeautoregressiveprocessoforder pby
X(t)=e Z(t), t 0, (3)
⊤1 ≥
where x means the transpose of x Rp. We say that X(t) is a CAR(p)-
⊤ t 0
process.Forq Nwith p>q,wedefi∈neaCARMA(p,q)-{proce}ss≥by
∈
X(t)=b Z(t), t 0, (4)
⊤
≥
whereb Rpisthevectorb=(b ,b ,...,b ,1,0,..,0) Rp,whereb =1and
0 1 q 1 ⊤ q
b =0,i∈=q+1,...,p 1. Note that b=e−yields a CAR∈(p)-process. Using the
i 1
−
Euler-Maryuamaapproximationscheme, the CARMA(p,q)-process X(t) on
t 0
a discretized time grid can be related to an autoregressive moving{averag}e≥time
series process of order p,q (see [8, Eq. (4.17)]). An explicit dynamics of the
CARMA(p,q)-process X(t) is(seee.g.[9,Lemma10.1])
t 0
{ }≥
t
X(t)=b exp(tC )Z + b exp((t s)C )e dL(s), (5)
⊤ p 0 ⊤ p p
0 −
Z
whereexp(tC )isthematrixexponentialoftC ,thematrixC multipliedbytimet.
p p p
1 Theoddlabellingoftheseconstants isexplained bytherelationship withautoregressive time
series,wherea iisrelated(butnotone-to-one)totheregressionoflagi,i=1,...,p.
Hilbert-valuedCARMAprocesses 3
IfC has only eigenvalueswith negativerealpart, then the process X admits a
p
limitingdistributionm withcharacteristicexponent(see[11])
X
¥
m (z):= limlogE eizX(t) = y b exp(sC )e z ds.
X t ¥ 0 L ⊤ p p
→ h i Z (cid:16) (cid:17)
Here,Y istbhelog-characteristicfunctionofL(1)(seee.g.[2])andlogthedistin-
L
guishedlogarithm(seee.g.[19,Lemma7.6]).Inparticular,ifL=s B withs >0
constantandBastandardBrownianmotion,wefind
1 ¥
m X(z)= z2s 2 (b⊤exp(sCp)ep)2ds,
−2 0
Z
and thus X has a Gabussian limiting distribution m X with zero mean and variance
s 2 ¥ (b exp(sC )e )2ds.
0 ⊤ p p
WhenX admitsalimitingdistribution,wehaveastationaryrepresentationofthe
R
processX suchthatX(t) m forallt R,namely,
X
∼ ∈
t
X(t)= b exp((t s)C )e dL(s), (6)
⊤ p p
¥ −
Z−
whereLisnowatwo-sidedLe´vyprocess.ThislinksCARMA(p,q)-processestothe
moregeneralclassofLe´vysemistationary(LSS)processes,definedin[4]as
t
X(t):= g(t s)s (s)dL(s), (7)
¥ −
Z−
forg:R Rbeingameasurablefunctionands apredictableprocesssuchthat
+
s g(t s→)s (s)fors t isintegrablewithrespecttoL.Indeed,LSSprocessesare
7→ − ≤
againaspecialcaseofso-calledambitfields,whicharespatio-temporalstochastic
processesoriginallydevelopedin[5]formodellingturbulence.Infact,theinfinite
dimensionalCARMAprocessesthatwearegoingtodefineinthispaperwillforma
subclassofambitfields,aswewillseeinSection4.WenotethatCARMAprocesses
withvaluesinRnhavebeendefinedandanalysedby[16],[20]andrecentlyin[14].
2 Definition ofCARMA processes inHilbertspace
Given p N, and let H for i = 1,...,p be separable Hilbert spaces with inner
i
∈
productsdenotedby , and associated norms . We define the productspace
i i
h· ·i |·|
H :=H ... H , which is again a separable Hilbert space equipped with the
1 p
inner pro×duct ×x,y :=(cid:229) p x,y and the induced norm denoted =(cid:229) n
for x=(x ,...h,x )i,y=(iy=1,h..i.,yiii) H. The projectionoperator P|·|:H i=1H|·i|si
1 p 1 p i i
definedas Px=x forx H,i=1∈,...,p. Itis straightforwardto see tha→tits ad-
i i
joint P :H H is give∈n by P x=(0,...,0,x,0,...,0) for x H, where the
i∗ i → i∗ ∈ i
x appearsin the ith coordinateof the vector consisting of p elements. IfU andV
4 FredEspenBenthandAndre´Su¨ss
aretwoseparableHilbertspaces,wedenoteL(U,V)theBanachspaceofbounded
linearoperatorsfromU toV,equippedwiththeoperatornorm .TheHilbert-
op
k·k
SchmidtnormforoperatorsinL(U,V)isdenoted ,andL (U,V)denotesthe
HS 2
k·k
spaceofHilbert-Schmidtoperators.IfU =V,wesimplywriteL(U)forL(U,U).
LetA :H H ,i=1,...,pbe p(unbounded)denselydefinedlinearoper-
i p+1 i p
ators,andI :H− → H ,i=2,...,pbeanother p 1(unbounded)densely
i p+2 i p+1 i
defined linear oper−ato→rs. Defi−ne the linear operator C :−H H represented as a
p
→
p pmatrixofoperators
×
0 I 0 . .. 0
p
0 0 I 0.. 0
p 1
 − 
. . . . .. .
C = . (8)
p  . . . . .. . 
 
 0 . . . .. I2 
 
Ap Ap 1 . . .. A1
 − 
 
SincetheA’sandI’saredenselydefined,C hasdomain
i i p
Dom(C )=Dom(A ) (Dom(A ) Dom(I )) ... (Dom(A ) Dom(I )),
p p p 1 p 1 2
× − ∩ × × ∩
which we suppose is dense in H. We note in passing that typically, H =H =
1 2
...=H and I =Id, the identity operator on H, i=1,...,p. Then Dom(C )=
p i i p
Dom(A ) Dom(A ) ... Dom(A ),whichisdenseinH.
p p 1 1
Onaco×mpletepro−bab×ility×space(W ,F,P)withfiltrationF = F satisfy-
t t 0
ingthe usualhypotheses,denotebyL:= L(t) a zero-means{quar}e-≥integrable
t 0
H -valuedLe´vyprocesswithcovarianceo{perato}r≥Q(i.e.,asymmetricnon-negative
p
definitetraceclassoperator),seee.g.[18,Sect.4.9].Considerthefollowingstochas-
ticdifferentialequation.Fort 0,
≥
dZ(t)=C Z(t)dt+P dL(t), Z(0):=Z H. (9)
p p∗ 0∈
ThenextpropositionstatesanexplicitexpressionforZ:= Z(t) :
t 0
{ }≥
Proposition1.Assume that C defined in (8) is the generator of aC -semigroup
p 0
S (t) onH.ThentheH-valuedstochasticprocessZgivenby
p t 0
{ }≥
t
Z(t)=S (t)Z + S (t s)P dL(s)
p 0 0 p − p∗
Z
istheuniquemildsolutionof (9).
Proof. WehavethatS (t s)P L(H ,H),andmoreover,since P =1it
p − p∗∈ p k p∗kop
follows
S (t s)P Q1/2 S (t s) P Q1/2 Kec(t s) Q1/2
k p − p∗ kHS≤k p − kopk p∗kopk kHS≤ − k kHS
bythe generalexponentialgrowthboundon the operatornormofaC -semigroup
0
(seee.g.[12,Prop.I.5.5]).Thus,forallt 0,
≥
Hilbert-valuedCARMAprocesses 5
t K
0 kSp(t−s)Pp∗Q1/2k2HSds≤ 2ce2ctkQ1/2k2HS<¥
Z
becauseQistraceclassbyassumption.ThestochasticintegralwithrespecttoLin
thedefinitionofZisthereforewell-defined.Hence,theresultfollowsdirectlyfrom
thedefinitionofmildsolutionsin[18,Def.9.5].
FromnowonwerestrictourattentiontooperatorsC in(8)whichadmitaC -
p 0
semigroup S (t) .We remarkinpassingthatinthenextsectionwewillpro-
p t 0
videarecur{sivede}fin≥itionofthesemigroup S (t) inaspecialsituationwhere
p t 0
allinvolvedoperatorsareboundedexceptA{. }≥
1
ACARMAprocesswithvaluesinaHilbertspaceisdefinednext:
Definition1. Let U be a separable Hilbert space. For L L(H,U), define the
U
∈
U-valuedstochasticprocessX := X(t) by
t 0
{ }≥
X(t):=L Z(t),t 0,
U
≥
forZ(t)definedin(9).Wecall X(t) aCARMA(p,U,L )-process.
t 0 U
{ }≥
Note thatwe do nothaveany q-parameterpresentin the definition,as in the real-
valuedcase(recall(4)).Insteadwe introducea Hilbertspaceandalinearoperator
as the ”second” parameters in the CARMA(p,U,L )-process. Indeed, the vector
U
binthereal-valuedCARMA(p,q)-processdefinedin(4)canbeviewedasalinear
operator from Rp into R by the scalar product operation Rp z bz R, or,
′
bychoosingU =Hi=R,LRz=b′z. Thisalso demonstratest∋hata7→nyrea∈l-valued
CARMA(p,q)-processisaCARMA(p,R,b )-processaccordingtoDefinition1.
′
·
FromProposition1wefindthattheexplicitrepresentationof X(t) is
t 0
{ }≥
t
X(t)=L S (t)Z + L S (t s)P dL(s), (10)
U p 0 0 U p − p∗
Z
fort 0.Notethatbylinearityofthestochasticintegralwecanmovetheoperator
L i≥nside.Furthermore,thestochasticintegraliswell-definedsinceL L(H,U)
U U
∈
andthushasafiniteoperatornorm.
Thecontinuous-timeautoregressive(CAR)processesconstituteaparticularlyin-
terestingsubclassoftheCARMA(p,U,L )-processes:
U
Definition2. TheCARMA(p,H ,P )-process X(t) fromDefinition1iscalled
1 1 t 0
anH -valuedCAR(p)-process. { }≥
1
TheexplicitdynamicsofanH -valuedCAR(p)-processbecomes
1
t
X(t)=P1Sp(t)Z0+ 0 P1Sp(t−s)Pp∗dL(s), (11)
Z
for t 0. In this paper we will be particularly focused on H -valued CAR(p)-
1
≥
processes.
6 FredEspenBenthandAndre´Su¨ss
Remark that the process L:=P L defines an H-valued Le´vy process which
p∗
hasmeanzeroandissquareintegrable.Itscovarianceoperatoriseasilyseentobe
P QP .
p∗ p
ItisimmediatetoseethatanH -valuedCAR(1)processisanOrnstein-Uhlenbeck
1
processdefinedonH ,with
1
dX(t)=A X(t)dt+dL(t),
1
andthus
t
X(t)=S (t)Z + S (t s)dL(s),
1 0 1
0 −
Z
beingitsmildsolution.
AnH -valuedCAR(p)processforp>1canbeviewedasahigher-order(indeed,
1
apthorder)stochasticdifferentialequation,aswenowdiscuss.Tothisend,suppose
thatRan(A ) Dom(I )andRan(I ) Dom(I ),andassumethatthereexistp 1
q 2 q q+1
⊂ ⊂ −
linear(unbounded)operatorsB ,B ,...,B suchthat
1 2 p 1
−
I I A =B I I I . (12)
p 2 q q p p 1 q+1
··· − ···
forq=1,...,p 1.WenotethatI I :H H andhenceI I A :H
p 2 p 1 p 2 q p+1 q
H . Moreover,−I I : H ··· H , a→nd therefore B : H··· H . We−su→p-
1 p q+1 p+1 q 1 q 1 1
posethatDom(B ·)·i·ssothatDo−m(B→I I I )=Dom(A ),a→ndwenotethat
q q p p 1 q+1 q
Dom(A )isthedomainoftheoperatorI − ·I··A .Forcompleteness,wedefinethe
q p 2 q
···
operatorB :H H as
p 1 1
→
B :=I I A . (13)
p p 2 p
···
Weseethatthisdefinitionisconsistentwiththeinductiverelationsin(12)(wesup-
pose that B is a linear (possibly unbounded) operator with domain Dom(B )=
p p
Dom(A )). With this at hand, we introducethe operator-valued pth-orderpolyno-
p
mialQ (l )forl C,
p
∈
Qp(l )=l p B1l p−1 B2l p−2 Bp 1l Bp. (14)
− − −···− − −
By definition, X(t)=P Z(t), which is the first coordinate in the vector Z(t)=
1
(Z (t),...,Z (t)) H.From(9)andthedefinitionoftheoperatormatrixC in(8),
1 p ⊤ p
∈
wefindthatZ (t)=I Z (t),Z (t)=I Z (t),...,Z (t)=I Z (t)andfinally
1′ p 2 2′ p 1 3 ′p 1 2 p
− −
Z (t)=A Z (t)+ A Z (t)+L˙(t).
′p p 1 ··· 1 1
Here, L˙(t) is the formal time derivative of L. By iteration, we find that Z(q)(t)=
1
I I I Z (t)forq=1,...,p 1.Thus,
p p−1··· p−(q−1) q+1 −
Hilbert-valuedCARMAprocesses 7
d
(p) (p 1)
Z1 = dtZ1 − =Ip···I2Z′p(t)
=I I A Z (t)+I I A Z (t)+ +I I A Z (t)+I I L˙(t)
p 2 p 1 p 2 p 1 2 p 2 1 p p 2
··· ··· − ··· ··· ···
=BpZ1(t)+Bp−1Z1′(t)+Bp−2Z1(2)(t)+...+B1Z1(p−1)(t)+Ip···I2L˙(t).
Inthelastequalitywemadeuseof(12)and(13).WefindthatanH -valuedCAR(p)
1
processX(t)=P Z(t)canbeviewedasasolutiontothe pth-orderstochasticdif-
1
ferentialequationformallyexpressedby
d
Q X(t)=I I L˙(t). (15)
p p 2
dt ···
(cid:18) (cid:19)
Re-expressingEq.15wefindthestochasticdifferentialequation
p
dX(p 1)(t)= (cid:229) B X(p q)(t) dt+I I dL(t). (16)
− q − p 2
q=1 ! ···
IfH =...=H andC isaboundedoperator,thenB =I I A in(12)whenever
1 p p q q 2 q
···
I I A commutes with I I . In this sense the condition (12) is a specific
q 2 q p q+1
··· ···
commutationrelationship on A and the operatorsI ,...,I . In the particular case
q 2 p
I =Idfori=2,...,p,thenwetriviallyhaveA =B forq=1,...,p.
i q q
Weendthissectionwithshowingthatthestochasticwaveequationcanbeviewed
asanexampleofaHilbert-valuedCAR(2)-process.Tothisend,letH :=L2(0,1),
2
the space of square-integrable functions on the unit interval, and consider the
stochasticpartialdifferentialequation
¶ 2Y(t,x) ¶ 2Y(t,x) ¶ L(t,x)
= + , (17)
¶ t2 ¶ x2 ¶ t
witht 0andx (0,1).Wecanrephrasethiswaveequationas
≥ ∈
Y(t,x) 0 Id Y(t,x) 0
d ¶Y(t,x) = D 0 ¶Y(t,x) dt+ dL(t,x) , (18)
(cid:20) ¶ t (cid:21) (cid:20) (cid:21)(cid:20) ¶ t (cid:21) (cid:20) (cid:21)
withD =¶ 2/¶ x2beingtheLaplaceoperator.Theeigenvectorse (x):=√2sin(p nx),
n
n N,forD formanorthonormalbasisofL2(0,1).IntroducetheHilbertspaceH
1
as∈thesubspaceofL2(0,1)forwhich f 2:=p 2(cid:229) ¥ n2 f,e 2<¥ .FollowingEx-
| |1 n=1 h ni2
ampleB.13in[18],
0 Id
C =
2 D 0
(cid:20) (cid:21)
generatesaC -semigroupS (t)onH:=H H .TheLaplaceoperatorD isaself-
0 2 1 2
adjointnegativedefinite operatoronH . The×semigroupS (t) canbe represented
1 2
as
cos(( D )1/2t) ( D ) 1/2sin(( D )1/2t)
S2(t)= ( D )1/2−sin(( D )1/2t) − c−os(( D )1−/2t) . (19)
(cid:20)− − − − (cid:21)
8 FredEspenBenthandAndre´Su¨ss
In the previousequality,we definefor a real-valuedfunctiong the linear operator
g(D ) using functional calculus, i.e., g(D )f =(cid:229) ¥ g( p 2n2) f,e e whenever
n=1 − h ni2 n
thissumconverges.Theseconsiderationsshowthatthewaveequationisaspecific
exampleofaCAR(2)-process.
3 AnalysisofCARMA processes
In this section we derive some fundamental properties of CARMA processes in
Hilbertspaces.
3.1 Distributionalproperties
We state the conditionalcharacteristic functionalof a CARMA(p,U,L )-process
U
inthenextproposition.
Proposition2.AssumeX isaCARMA(p,U,L )-process.Then,forx U,
U
∈
t s
E eiX(t),xU F =exp i L S (t)Z ,x + − y P S (u)L x du
h i | s h U p 0 iU 0 L p p∗ U∗
h i (cid:18) s Z (cid:0) (cid:1) (cid:19)
exp i L S (t u)P dL(u),x ,
× h 0 U p − p∗ iU
(cid:18) Z (cid:19)
for0 s t.Here,y isthecharacteristicexponentoftheLe´vyprocessL.
L
≤ ≤
Proof. From(10)itholdsfor0 s t,
≤ ≤
s t
X(t)=L S (t)Z + L S (t u)P dL(u)+ L S (t u)P dL(u).
U p 0 0 U p − p∗ s U p − p∗
Z Z
The Le´vy process has independent increments, and F -measurability of the first
s
stochasticintegralthusyields
s
E eihX(t),xiU|Fs =exp ihLUSp(t)Z0,xiU+ih 0 LUSp(t−u)Pp∗dL(u),xiU
h i (cid:18) Z (cid:19)
t
E exp i L S (t u)P dL(u),x .
× h s U p − p∗ iU
(cid:20) (cid:18) Z (cid:19)(cid:21)
Theresultfollowsfrom[18,Chapter4].
Suppose now that L=W, an H -valued Wiener process. Then the characteristic
p
exponentis
1
y (h)= Qh,h ,
W p
−2h i
Hilbert-valuedCARMAprocesses 9
forh H .Hence,fromProp.2itfollowsthat,
p
∈
s
E eiX(t),xU F =exp i L S (t)Z ,x +i L S (t u)P dW(u),x
h i | s h U p 0 iU h 0 U p − p∗ iU
h i (cid:18) Z (cid:19)
1 t s
exp − L S (u)P QP S (u)L x,x du
× −2 0 h U p p∗ p p∗ U∗ iU
(cid:18) Z (cid:19)
WefindthatX(t)F fors t isaGaussianprocessinH ,withmean
s 1
| ≤
s
E[X(t) F ]=L S (t)Z + W S (t u)P dL(u)
| s U p 0 0 U p − p∗
Z
andcovarianceoperator
t s
Var(X(t)F )= − L S (u)P QP S (u)L du,
| s 0 U p p∗ p p∗ U∗
Z
wheretheintegralisinterpretedintheBochnersense.IfthesemigroupS (u)isex-
p
ponentiallystable,thenX(t)F admitsaGaussianlimitingdistributionwithmean
s
|
zeroandcovarianceoperator
¥
tlim¥ Var(X(t)|Fs)= 0 LUSp(u)Pp∗QPpSp∗(u)LU∗du.
→ Z
This is the invariantmeasure of X. We remark in passing that measures on H are
definedonitsBorels -algebra.
In[3]thereisananalysisofinvariantmeasuresofLe´vy-drivenOrnstein-Uhlenbeck
processes. We discuss this here in the context of the Ornstein-Uhlenbeck process
Z(t) defined in (9). Assume m is the invariant measure of Z(t) , and
t 0 Z t 0
{recall}th≥edefinitionofitscharacteristicexponentm (x), { }≥
Z
m Z(x)=logE eihx,Z(bt)i . (20)
h i
Here, x H and log is the dbistinguished logarithm(see e.g. [19, Lemma 7.6]). If
Z m ∈,then,indistribution,Z =Z(t)forallt 0anditfollowsthatthecharac-
0 Z 0
teri∼sticexponentofm satisfies, ≥
Z
t
m (x)=m (S (t)x)+ y (P S (u)x)du (21)
Z Z p∗ L p p∗
0
Z
foranyx H andbt 0.Folblowing[3],m Zbecomesanoperatorself-decomposable
∈ ≥
distributionsince,
m =S (t)m ⋆m . (22)
Z p Z t
Here, m is the distribution of tS (u)P dL(u), ⋆ is the convolution product of
t 0 p p∗
measuresandS (t)m :=m S (t) 1isaprobabilitydistributiononH,givenby
p Z Z◦R p −
10 FredEspenBenthandAndre´Su¨ss
f(x)(S (t)m )(dx)= f(S (t) x)m (dx),
p Z p ∗ Z
H H
Z Z
foranyboundedmeasurablefunction f :H R.IfZ(t) m ,thensince
Z
→ ∼
logE eihLUZ(t),xiU =logE ehZ(t),LU∗xi =m Z(LU∗x),
h i h i
itfollowsthat X(t) t 0isstationarywithdistributionm Xbhavingcharacteristicex-
ponentm (x)={ m (}L≥x)forx U.
WenoXticethatZC iUs∗abound∈edoperatoronH ifandonlyifA,i=1,...,pand
p i
Ij,j=2b,...,p arbe boundedoperators.In the case of Cp being bounded,we know
from Thm.I.3.14in [12] that the semigroup S (t) is exponentiallystable if
p t 0
and only if Re(l )<0 for all l s (C ), w{here s (}C≥) denotes the spectrum of
p p
theboundedoperatorC .Recallfr∈omSection1thatareal-valuedCARMAprocess
p
admitsalimitingdistributionifandonlyifalltheeigenvaluesofC inEq.2have
p
negativerealpart.Ingeneral,byThm.V.1.11in[12],thesemigroup S (t) is
p t 0
exponentiallystableifandonlyif l C Re(l )>0 isasubsetof{theres}ol≥vent
sCetrfo(rClp)orf(CCp a)n.dsupRe(l )>0kR({l ,∈Cp)k|<¥ .Here},R(l ,Cp)istheresolventof
p p
∈
3.2 Semimartingalerepresentation
Letusstudya semimartingalerepresentationof the CAR(p) process.We havethe
followingproposition:
Proposition3.For p Nwith p>1,assumethatC definedin(8)isthegenera-
p
torofaC -semigroup∈S (t) .ThentheH -valuedCAR(p)processX givenin
0 p t 0 1
Definition2hastherep{resenta}ti≥on
t u
X(t)=P S (t)Z +P C S (u s)P dL(s)du,
1 p 0 1 p 0 0 p − p∗
Z Z
forallt 0.
≥
Proof. From[12,Ch.II,Lemma1.3],wehavethat
t
S (t)=Id+C S (s)ds.
p p p
0
Z
Butforanyx H,itissimpletoseethatP IdP x=0when p>1.Thereforeit
∈ 1 p∗
holds
t
P S (t)P =P C S (s)P ds.
1 p p∗ 1 p p p∗
0
Z
Theintegralontheright-handsideisinBochnersenseasanintegralofoperators.
AfterappealingtothestochasticFubinitheorem,itfollowsfromtheexplicitexpres-
sionofX(t)in(11)