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Continuous-time autoregressive moving-average processes in Hilbert space FredEspenBenthandAndre´Su¨ss 7 1 0 2 n a J 7 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)