Multivariate Generalized Ornstein-Uhlenbeck Processes

Transcription

1 Mulivariae Generalized Ornsein-Uhlenbeck Processes Ania Behme and Alexander Lindner Absrac De Haan and Karandikar [12] inroduced generalized Ornsein Uhlenbeck processes as one-dimensional processes (V 0 which are basically characerized by he fac ha for each h > 0 he equidisanly sampled process (V nh n N0 saisfies he random recurrence equaion V nh = A (nh,nh V (nh + B (nh,nh, n N, where (A (nh,nh, B (nh,nh n N is an i.i.d. sequence wih posiive A 0,h for each h > 0. We generalize his concep o a mulivariae seing and use i o define mulivariae generalized Ornsein Uhlenbeck (MGOU processes which occur o be characerized by a saring random variable and some Lévy process (X, Y in R m m R m. The sochasic differenial equaion an MGOU process saisfies is also derived. We furher sudy invarian subspaces and irreducibiliy of he models generaed by MGOU processes and use his o give necessary and sufficien condiions for he exisence of sricly saionary soluions of MGOU processes under some exra condiions Mahemaics subjec classificaion. 60G10, 60G51. Key words and phrases. generalized Ornsein-Uhlenbeck process, invarian subspace, irreducible model, Lévy process, muliplicaive Lévy process, sochasic exponenial. 1 Inroducion Le (ξ, η = (ξ, η 0 be a bivariae Lévy process and V 0 a random variable, independen of (ξ, η. Then, following De Haan and Karandikar [12] and Carmona e al. [6], he onedimensional process (V 0, given by V = e (V ξ 0 + e ξ s dη s, 0, (1.1 Technische Universiä Braunschweig, Insiu für Mahemaische Sochasik, Pockelssraße 14, D Braunschweig, Germany, a.behme@u-bs.de or a.lindner@u-bs.de, el.:+49/531/ or +49/531/ , fax:+49/531/

2 is called a generalized Ornsein Uhlenbeck (GOU process. We refer o Maller e al. [19] for furher informaion and references regarding GOU processes. A key feaure of hese processes is ha for any h > 0, he random sequence (V nh n N0 saisfies he random recurrence equaion V nh = A (nh,nh V (nh + B (nh,nh, n N, where (A (nh,nh, B (nh,nh n N is an i.i.d. (independen and idenically disribued sequence wih A 0,h > 0 almos surely. Wihou assuming independence of V 0 and (ξ, η, processes of he form (1.1 are he only processes having his propery for any h > 0 and which saisfy some naural exra condiions, as shown by De Haan and Karandikar [12]. In he presen paper we exend he seing of De Haan and Karandikar [12] o random marices wih real valued enries, i.e. we aim o consruc a process (V 0, wih V = (V (i,j 1 i m R m l 1 j l in coninuous ime which fulfills he random recurrence equaion V = A s, V s + B s, a.s., 0 s, (1.2 for random funcionals (A s, 0 s, (B s, 0 s such ha A s, R m m and B s, R m l, he A s, are supposed o be non-singular and (A (nh,nh, B (nh,nh, n N, are i.i.d. for all h > 0. We also aim o characerize all processes in coninuous ime which have his propery and saisfy some naural exra condiions. The obained soluions will be called mulivariae generalized Ornsein Uhlenbeck (MGOU processes since hey exend he key feaure of one-dimensional generalized Ornsein Uhlenbeck processes canonically. Observe ha he quesion of when a soluion of (1.2 exiss can be reaed separaely for each column of (V 0. Thus, if no saed oherwise, for simpliciy we se l = 1 hroughou his paper, hence V and B s, are elemens in R m. To moivae he menioned exra condiions, following he lines of De Haan and Karandikar [12] observe ha he condiion of (1.2 o hold for all 0 s yields A u, V u + B u, = V = A s, V s + B s, = A s, A u,s V u + A s, B u,s + B s,, 0 u s. Assuming ha (A s,, B s, 0 s is unique now leads o Assumpion 1(a given below while exending he i.i.d. propery of (A (nh,nh, B (nh,nh, n N, for all h > 0 ino he coninuous ime seing yields he requiremens 1(b and (c. Finally, i is naural o impose ha (A 0, 0 and (B 0, 0 are coninuous in probabiliy a 0 since his, ogeher wih 1(a,(b and (c, implies he exisence of càdlàg modificaions of he processes (A 0 := (A 0, 0 and (B 0 := (B 0, 0 as will be shown in Lemma 2.1 below. This moivaes Assumpion 1(d below. We denoe he se of all inverible real m m-marices by GL(R, m, he ideniy marix by I and by 0 he vecor (or marix having only zero enries. We wrie d = for equaliy in disribuion and P- lim for limis in probabiliy. Assumpion 1. Suppose he GL(R, m R m -valued random funcional (A s,, B s, 0 s wih A, = I and B, = 0 a.s. for all 0 saisfies he following four condiions. 2

3 (a For all 0 u s almos surely A u, = A s, A u,s and B u, = A s, B u,s + B s,. (1.3 (b For all 0 a b c d he families of random marices {(A s,, B s,, a s b} and {(A s,, B s,, c s d} are independen. (c For all 0 s i holds (A s,, B s, d = (A 0, s, B 0, s. (1.4 (d I holds P- lim 0 A 0, = I and P- lim 0 B 0, = 0. (1.5 The firs main resul of he paper will be a characerizaion of all random funcionals (A s,, B s, 0 s which saisfy Assumpion 1, in erms of appropriae driving Lévy processes. This will be achieved in Theorem 3.1 and hen be used o define MGOU processes as processes which saisfy (1.2 wih (A s,, B s, 0 s subjec o Assumpion 1. I will be also shown in Secion 3 ha MGOU processes saisfy he sochasic differenial equaion (SDE dv = du V + dl for appropriae Lévy processes U and L if he saring random variable V 0 is independen of (A 0,, B 0, 0, exending a corresponding one-dimensional resul of De Haan and Karandikar [12]. A new aspec compared o he one-dimensional GOU process is he possibiliy of he exisence of affine subspaces H of R m which are invarian under he model (1.2 in he sense ha A s, H + B s, H holds for all 0 s. In Secion 4 we give necessary and sufficien condiions for he exisence of an invarian affine subspace of he model (1.2 and show ha given he exisence of a d-dimensional invarian affine subspace H, afer an appropriae orhogonal ransformaion of he underlying space, he MGOU process wih V 0 H consiss of an (m d-dimensional consan process and an R d -valued MGOU process. Subsequenly in Secion 5 sricly saionary soluions of MGOU processes are reaed. Under some exra condiions we give necessary and sufficien condiions for heir exisence and deermine heir form, exending corresponding one-dimensional resuls of Behme e al. [3] and Lindner and Maller [18]. The proofs for he resuls of Secions 3 5 are given in Secions 6 8. A crucial ingredien for he derivaion of he necessary and sufficien condiions for saionariy are he resuls on saionary soluions of random recurrence equaions by Bougerol and Picard [4]. Secion 8 also conains several auxiliary resuls abou mulivariae sochasic exponenials. Some preliminary resuls are colleced in Secion 2, where we also se furher noaion used hroughou he paper. Random recurrence equaions have many applicaions in finance, biology or fracal images, o name jus a few, see e.g. Wong and Li [27], Tong [26], or Diaconis and Freedman [7]. Hence mulivariae generalized Ornsein Uhlenbeck processes as heir coninuous ime counerpars have considerable poenial for applicaions. In one dimension, various applicaions of he GOU process are known. For example, he volailiy of he COGARCH(1,1 process of Klüppelberg e al. [16] or he risk process of Paulsen [20] are one-dimensional GOU processes. As an example of an applicaion of he MGOU process 3

4 o finance, we presen in Example 3.6 he sae vecor process of he volailiy process of he COGARCH(q, p model of Brockwell e al. [5] as a special case of an MGOU process. Furher applicaions of MGOU processes as mulivariae volailiy models seem possible, bu we shall no pursue his opic furher in his paper bu leave i o fuure research. Finally, we menion ha major pars of he resuls of his paper have been obained in he firs named auhor s docoral hesis [2, Chaper 5]. 2 Preliminaries Throughou his paper for any marix M R m n we wrie M for is ranspose and le M (i,j denoe he componen in he ih row and jh column of M. Limis in disribuion will be denoed by d- lim or d, limis in probabiliy by P- lim or P, and almos surely will be abbreviaed by a.s.. The law of a random marix Y will be denoed by L(Y. We wrie N = {1, 2,...}, N 0 = {0, 1, 2,...} and log + (x := log max{x, 1} for x R. Jumps of a marix valued càdlàg process X = (X 0 will be denoed by X := X X wih X := lim s X s for > 0 and he convenion X 0 := 0. Muliplicaive Lévy processes Recall ha an (addiive Lévy process X = (X 0 wih values in R m l is a process wih saionary and independen (addiive incremens which has almos surely càdlàg pahs and sars a 0. Here, an incremen of X is given by X X s for s. We refer o Applebaum [1] or Sao [23] for furher informaion regarding Lévy processes. In he following i will be also necessary o consider muliplicaive Lévy processes wih values in he general linear group GL(R, m of order m, where he group operaion is marix muliplicaion. For ha, remark ha he group srucure allows us o define (muliplicaive lef incremens X Xs and (muliplicaive righ incremens Xs X for 0 s < of a GL(R, m-valued process. We say ha he process (X 0 in GL(R, m has independen lef incremens if for any n N, 0 < 1 <... < n, he random variables X 0, X 1 X0,..., X n X n are independen. The process has saionary lef incremens if X Xs d = X s X0 holds for all s <. Saionariy and independence of righ incremens is undersood analogously. Now following he noaions in he book of Liao [17] a càdlàg process (X 0 in GL(R, m, m 1, wih X 0 = I a.s. is called a (muliplicaive lef Lévy process, if i has independen and saionary righ incremens. Similarly, a càdlàg process (X 0 in GL(R, m, m 1, wih X 0 = I a.s. is called a (muliplicaive righ Lévy process, if i has independen and saionary lef incremens. Given a filraion F = (F 0, a lef Lévy process (X 0 in GL(R, m is called a lef F-Lévy process, if i is adaped o F and for any s < he righ incremen Xs X is independen of F s. Righ F-Lévy processes and (addiive F-Lévy processes are defined similarly. The following lemma gives he connecion beween he random funcionals A s, saisfying Assumpion 1 and muliplicaive Lévy processes. 4

5 Lemma 2.1. (a For any (A s, 0 s fulfilling Assumpion 1 he process (A 0 = (A 0, 0 has a càdlàg modificaion which is a righ Lévy process in GL(R, m. Conversely, if (A 0 is a righ Lévy process in GL(R, m, hen (A s, 0 s defined by A s, = A A s fulfills Assumpion 1. (b For any (A s,, B s, 0 s fulfilling Assumpion 1 he process (A, B 0 = (A 0,, B 0, 0 has a càdlàg modificaion. Proof. (a Since by Assumpion 1(a we have A A s = A s, i follows direcly from Assumpion 1(b and (c, ha (A 0 is a sochasic process in GL(R, m wih saionary and independen lef incremens. I is everywhere coninuous in probabiliy from he righ since by 1(a, (c and (d Similarly due o P- lim h 0 A +h = P- lim h 0 A,+h A = A, 0. P- lim h 0 A h = P- lim h 0 A A h, = A P- lim h 0 A h = A, 0, i is also coninuous in probabiliy from he lef such ha by [25, Theorem V.3] a càdlàg modificaion exiss which is a righ Lévy process in GL(R, m as specified above. The converse is rue by he definiion of righ Lévy processes. (b Since B +h = A +h A B + B,+h he process (B 0 is by Assumpion 1(c and (d everywhere coninuous in probabiliy from he righ and similarly from he lef. Hence i admis a càdlàg modificaion which can be shown by a simple exension of he proof in he one-dimensional case given in [12, Lemma 2.1]. Since every se (A s,, B s, 0 s of random funcionals saisfying Assumpion 1 admis a càdlàg modificaion (A, B 0 by he preceding lemma, we may and do resric aenion o such funcionals wih càdlàg pahs. Marix valued sochasic inegrals Given a filraion F = (F 0 saisfying he usual hypoheses (cf. [21, p. 3], a marixvalued sochasic process M = (M 0 is called an F-semimaringale or simply a semimaringale if every componen (M (i,j 0 is a semimaringale wih respec o he filraion F. For a semimaringale M in R m n and a locally bounded predicable process H in R l m he R l n -valued (lef sochasic inegral I = HdM is given by I (i,j = m k=1 H (i,k dm (k,j and in he same way for M R l m, H R m n, he R l n -valued sochasic (righ inegral J = dmh is given by J (i,j = m k=1 H (k,j dm (i,k. Sochasic inegrals of he form HdM H for locally bounded predicable processes H and H are defined similarly in he obvious way. Given wo semimaringales M and N in R l m and R m n he quadraic variaion [M, N] in R l n is defined by is componens via [M, N] (i,j = m k=1 [M (i,k, N (k,j ]. Similarly is 5

6 coninuous par [M, N] c is given by ([M, N] c (i,j = m k=1 [M (i,k, N (k,j ] c. Wih hese noaions, for wo semimaringales M and N in R m m and wo locally bounded predicable processes G and H in R m m we have he following a.s. equaliies as saed e.g. in Karandikar [15] [ ] G s dm s, dn s H s = G s d[m, N] s H s, 0, (2.1 (0, ] (0, ] [ M, (0, ] G s dn s ] = [ dm s G s, N (0, ] ], 0, (2.2 and he inegraion by pars formula akes he form (MN = M s dn s + dm s N s + [M, N], 0. (2.3 The mulivariae sochasic exponenial Sochasic exponenials of R m m -valued Lévy processes will play a crucial rule in our consideraions. We firs recall he definiion of lef and righ sochasic exponenials from [21, p ]. Definiion 2.2. Le (X 0 be a semimaringale in R m m. Then is lef sochasic exponenial E (X is defined as he unique R m m -valued, adaped, càdlàg soluion of he inegral equaion Z = I + Z s dx s, 0, (2.4 while he unique adaped, càdlàg soluion of Z = I + dx s Z s, 0, (2.5 will be called righ sochasic exponenial and denoed by E (X. Boh E (X and E (X are semimaringales. Unforunaely, unlike for one-dimensional sochasic exponenials as e.g. in [21, Theorem II.37], no closed form expression is available for general mulivariae sochasic exponenials, which makes heir reamen more difficul. The SDE of he sochasic exponenial for processes wih values in arbirary Lie groups has been sudied by Esrade [10]. Remark ha replacing Z and X by heir ransposes in (2.4 leads o he SDE (2.5 and vice versa. Hence we have E (X = E (X. (2.6 As has been observed by Karandikar [15] a necessary and sufficien condiion for nonsingulariy of he lef sochasic exponenial of an R m m -valued process X a ime, is o 6

7 claim ha (I + X s is inverible for all 0 < s. Due o he above saed relaionship beween lef and righ exponenial his resul holds rue also for righ exponenials and hence any sochasic exponenial is inverible for all 0 if and only if de(i + X 0 for all 0. (2.7 For GL(R, m-valued semimaringales, he sochasic logarihm is defined as follows. Definiion 2.3. Le (Z 0 be a GL(R, m-valued semimaringale wih Z 0 = I. Then he lef sochasic logarihm Log Z and righ sochasic logarihm Log Z of Z are defined by Log (Z = Zs dz s, and Log (Z = dz s Zs, 0, (2.8 respecively. I is clear from he defining SDE dz = Z dx for lef sochasic exponenials ha if X is a semimaringale saisfying (2.7 wih X 0 = 0, hen Log E (X = X and X is he unique semimaringale Y saisfying Y 0 = 0 and E (Y = E (X. The same is rue for righ sochasic exponenials and righ sochasic logarihms. The following one-o-one relaion beween muliplicaive Lévy processes and sochasic exponenials of addiive Lévy processes is a key observaion for he invesigaions in his paper. Proposiion 2.4. Le F = (F 0 be a filraion saisfying he usual hypoheses. Then for every F-Lévy process (X 0 in R m m fulfilling (2.7, he sochasic exponenial Z = E (X (resp. Z = E (X is a lef (resp. righ F-Lévy process in GL(R, m. Conversely, if Z = (Z 0 is a lef (resp. righ F-Lévy process in GL(R, m, hen Z is an F-semimaringale and Log Z (resp. Log Z is an addiive Lévy process in R m m saisfying (2.7. Skech of Proof. The firs par follows by simple calculaions using he Markov propery of X, and we refer o [2, Prop. 5.5] for a complee proof. The converse has been observed by Holevo [13] as a conclusion of resuls by Skorokhod [24]. Acually, here i is only observed ha Z is a semimaringale wih respec o is augmened naural filraion, H say, and ha Log Z and Log Z, resp., are H-Lévy processes, bu i is easy o see ha hen Log Z and Log Z are even F-Lévy processes, and since E ( Log Z = Z and E ( Log Z = Z, resp., i follows ha Z is an F-semimaringale. Again we refer o [2, Prop. 5.5] for deailed calculaions. Since he inverse and he ranspose of a lef Lévy process in GL(R, m are righ Lévy processes and vice versa, for any addiive Lévy process (X 0 fulfilling (2.7 he process ( E (X 0 is a righ Lévy process and hence by he above proposiion i is he righ 7

8 sochasic exponenial of anoher Lévy process (U 0. In fac (see [15, Theorem 1] if (X 0 is a semimaringale such ha (2.7 is fulfilled, hen i holds wih E (X = [ E (U ] = E (U, 0 U := X + [X, X] c + 0<s ( (I + Xs I + X s, 0. (2.9 Remark ha i follows from (2.9 by sandard calculaions ha he processes U and X fulfill he relaion U = X [X, U], 0, (2.10 and ha if X is a Lévy process, hen so is U and vice versa. 3 Mulivariae Generalized Ornsein-Uhlenbeck Processes In his secion we will characerize all families of random funcionals (A s,, B s, 0 s saisfying Assumpion 1 and hen will use his o define mulivariae generalized Ornsein- Uhlenbeck processes. Furher, we show ha every mulivariae generalized Ornsein- Uhlenbeck process (V 0 is a soluion of he SDE dv = du V + dl for a suiable R m m R m -valued Lévy process (U, L. Conversely, provided ha V 0 is F 0 -measurable for some filraion F = (F 0 saisfying he usual hypoheses such ha he Lévy process (U, L is a semimaringale wih respec o F, he soluion o his SDE is a mulivariae generalized Ornsein Uhlenbeck process. The proofs for he resuls of his secion are given in Secion 6. The following heorem characerizes all choices of random funcionals (A s,, B s, 0 s fulfilling Assumpion 1. Recall ha A = A 0,, B = B 0, and ha by Lemma 2.1 we can resric o càdlàg versions of (A, B 0. Theorem 3.1. Suppose ha (A s,, B s, 0 s saisfies Assumpion 1 and ha (A 0 and (B 0 are chosen o be càdlàg. Then here is a unique Lévy process (X, Y in R m m R m such ha X saisfies (2.7 and such ha ( As, = E (X E (Xs B a.s., 0 s. (3.1 s, E (X (s,] E (Xu dy u The Lévy process (X, Y is given by ( ( X = Log A Y A u d(a u B u, 0, (3.2 8

9 where he inegral is defined as a sochasic inegral wih respec o he naural augmened filraion of (A, B 0, for which (A 0 and (B 0 are semimaringales. Conversely, if (X, Y is a Lévy process in R m m R m such ha X saisfies (2.7, hen (A s,, B s, 0 s defined by he righ hand side of (3.1 saisfies Assumpion 1. Since a mulivariae generalized Ornsein-Uhlenbeck process (V 0 was supposed o saisfy (1.2 wih (A s,, B s, 0 s saisfying Assumpion 1, Theorem 3.1 moivaes he following definiion. Definiion 3.2. Le (X, Y = (X, Y 0 be a Lévy process in R m m R m such ha X saisfies (2.7 and le V 0 be a random variable in R m. Then he R m -valued process (V 0, given by ( V := E (X V 0 + E (Xs dy s, 0, (3.3 will be called mulivariae generalized Ornsein-Uhlenbeck (MGOU process driven by (X, Y 0. The MGOU process will be called causal or non-anicipaive, if V 0 is independen of (X, Y, and sricly non-causal if V is independen of (X s, Y s 0 s< for all 0. I is easy o see ha an MGOU process indeed saisfies (1.2. Remark ha even for m = 1 Definiion 3.2 is generalizing he sandard definiion of a generalized Ornsein-Uhlenbeck process since we do no assume a priori ha V 0 is independen of (X, Y 0 and also he condiion of E(X o be sricly posiive is dropped. Neverheless i seems naural o us o include hese cases in he class of generalized Ornsein-Uhlenbeck processes. Observe ha any MGOU process wih saring random variable V 0 independen of (X, Y is a ime-homogeneous Markov process. Example 3.3. (a If X = Λ for some Λ R m m is a pure drif process hen E (X = E (X = e Λ and he MGOU process V = e (V Λ 0 + e Λs dy s, 0, driven by (X, Y is he usual mulivariae Ornsein Uhlenbeck ype process driven by Y as inroduced in [22]. (b If (X, Y = (diag(x (1,1,..., X (m,m, (Y (1,..., Y (m, i.e. if X is a Lévy process concenraed on he diagonal marices, and X saisfies condiion (2.7, hen E (X = E (X = diag(e(x (1,1,..., E(X (m,m, where E( denoes he usual one-dimensional sochasic exponenial, and he ih componen V (i of he MGOU process (V 0 driven by (X, Y saisfies V (i = E(X (i,i ( V (i 0 + E(X (i,i s dy s (i 0, 0, i = 1,..., m. 9

10 I follows ha V (i is a one-dimensional MGOU process driven by (X (i,i, Y (i. If addiionally X (i,i does no have jumps of size less han or equal o and if V (i 0 is independen of (X (i,i, Y (i, hen V (i is a GOU process. Observe ha in general componens of MGOU processes are no MGOU processes if X is no concenraed on he diagonal marices. An MGOU process can also be characerized by he sochasic differenial equaion i saisfies. Theorem 3.4. (a Le (X, Y be a Lévy process in R m m R m such ha (2.7 holds, and le (V 0 be he MGOU process driven by (X, Y wih saring random variable V 0. Le F = (F 0 be some filraion saisfying he usual hypoheses such ha (X, Y is a semimaringale wih respec o F and V 0 is F 0 -measurable. Then (V 0 solves he SDE dv = du V + dl, (3.4 where (U, L is he Lévy process in R m m R m wih U as defined in (2.9 and L given by L = Y + ( (I + Xs I Y s [X, Y ] c, 0. (3.5 The process U saisfies 0<s de(i + U 0 for all 0. (3.6 (b Conversely, if (U, L is a Lévy process in R m m R m such ha U saisfies (3.6, F = (F 0 is a filraion saisfying he usual hypoheses such ha (U, L is an F-semimaringale and V 0 is an R m -valued F 0 -measurable saring random variable, hen he soluion o (3.4 is an MGOU process driven by (X, Y, where (X, Y is he Lévy process defined by ( ( X Log ( E (U = Y L + [ Log (, 0, (3.7 E (U, L] and X saisfies (2.7. Observe ha under he naural assumpion ha V 0 is independen of (X, Y (i.e. for a causal MGOU process, he smalles filraion F which saisfies he usual hypoheses and is such ha V 0 is F 0 measurable and (X, Y is adaped o F is a filraion such ha X, Y, U and L are semimaringales wih respec o i (cf. Corollary 1 of Theorem VI.11 in [21], as required in he saemen of (a. A similar remark holds for (b if V 0 is independen of (U, L. In he following proposiion we sae some cross-relaions beween (X, Y and (U, L defined by (2.9 and (

11 Proposiion 3.5. Le (X, Y be a Lévy process in R m m R m such ha X saisfies (2.7 and le (U, L be defined by (2.9 and (3.5. Then L = Y + [U, Y ], 0, (3.8 and Y = L + [X, L], 0. (3.9 Finally, we show in he nex example ha he sae vecor of he COGARCH(q, m volailiy process is an m-dimensional MGOU process. Example 3.6. Le m, q N, q m, c 1,..., c m, d 0,..., d m R wih c m 0 and d q 0, d q =... = d m = 0. Denoe d d 1 C = , e = 0., d = d m 2 c m c m c m 2 c 1 1 d m wih C R m m, e, d R m, and le M be a one-dimensional Lévy process wih non-rival Lévy measure. Le β > 0. Then, as defined in [5], he COGARCH(q, m process, driven by M and wih parameers C, β and d has (righ-coninuous volailiy process (S 0 given by S = β + d V, 0, (3.10 where he sae vecor process V = (V 0 is he unique càdlàg soluion of he sochasic differenial equaion dv = CV d + es d[m, M] (d = CV d + e(β + d V d[m, M] (d, 0, (3.11 wih iniial value V 0, independen of (M 0. Here, [M, M] (d = 0<s ( M s 2 denoes he discree par of he quadraic variaion of M. If he process (S 0 is non-negaive almos surely, condiions for which are given in Secion 5 of [5], hen G = (G 0, defined by G 0 = 0, dg = S dm, is called a COGARCH(q, m process wih parameers C, d, β and driving Lévy process M. I follows from [5, Theorem 3.3] and is proof ha he sae vecor process (V 0 saisfies (1.2 wih random funcionals (A s,, B s, which saisfy Assumpion 1, so ha (V 0 is an MGOU process. Using he SDE (3.11 and Theorem 3.4, we ge anoher proof of his, observing ha dv = CV d + βed V d[m, M] (d + βe d[m, M] (d = (C d + βed d[m, M] (d V + βe d[m, M] (d = du V + dl, 11

12 where U = C + β[m, M] (d ed and L = β[m, M] (d e. (3.12 Since he jumps of [M, M] (d are non-negaive, i follows ha U saisfies condiion (3.6 and hence ha V is a causal MGOU process by Theorem MGOU Processes Carried by Affine Subspaces In his secion we will classify MGOU processes which are carried by affine subspaces of R m. To do ha, we inroduce he noion of irreducibiliy which we mainly adop from Bougerol and Picard [4] who sudied generalized auoregressive models in discree ime. The proofs for he resuls of his secion are given in Secion 7. Definiion 4.1. Suppose (X, Y 0 is a Lévy process in R m m R m such ha X saisfies (2.7 and define (A s,, B s, 0 s by (3.1. Then an affine subspace H of R m is called invarian under he auoregressive model (1.2 if A s, H + B s, H, almos surely, holds for all 0 s. If R m is he only invarian affine subspace, he model (1.2 is called irreducible. Obviously, by Assumpion 1(c, i is enough o require he above condiion for s = 0 and all 0. Remark ha he given definiion of invarian subspaces is more resricive han he one in [4], since e.g. seing Y = B = 0 and leing A be a roaion operaor wih angle 2π implies ha in he discree ime model V n = A n,n V n + B n,n, n N, every poin is a zero-dimensional invarian affine subspace, while only he roaion axis is invarian for all 0. Accordingly, irreducibiliy of he coninuous ime model does no direcly imply ha for all h > 0 he discree ime model V nh = A (nh,nh V (nh +B (nh,nh, n N, is irreducible in he sense of [4]. Bu we can show he following proposiion which saes ha a leas here is some h > 0 for which he corresponding discree ime model is irreducible. This will be an imporan ingredien when proving Theorems 5.3, 5.4 and 5.7 below on he exisence of sricly saionary soluions. Proposiion 4.2. Suppose (X, Y 0 is a Lévy process in R m m R m such ha X saisfies (2.7 and define (A s,, B s, 0 s by (3.1. Suppose ha he auoregressive model (1.2 is irreducible. Then here exiss h > 0 for which he discree-ime auoregressive model V nh = A (nh,nh V (nh + B (nh,nh, n N, (4.1 is irreducible in he sense ha here exiss no affine subspace H of R m, H R m, such ha for all n N, A (nh,nh H + B (nh,nh H almos surely. The nex heorem reas MGOU processes where he corresponding auoregressive model admis a d-dimensional invarian affine subspace H. I urns ou ha in his case we can spli up he process carried by H in a consan par and an R m d -valued MGOU process. For convenience we firs assume ha H is parallel o he axes. 12

13 Theorem 4.3. Suppose (V 0 is an MGOU process wih saring random variable V 0, driven by he Lévy process (X, Y 0 in R m m R m, where X fulfills (2.7, and le (A s,, B s, 0 s as defined in (3.1. (a Assume ha H = {(k 1,..., k d, h d+1,..., h m, h d+1,..., h m R} wih 1 d m and consans k 1,..., k d R is an invarian, affine subspace of( R m wih respec o K he model (1.2. Then, given ha V 0 H a.s., i holds V = H a.s. for each 0 wih K = (k 1,..., k d and V R m d, and he Lévy processes X and Y saisfy for all 0 ( X 1 X = 0 X 2 X 3 a.s. where X 1 R d d and (4.2 ( ( Y 1 Y = X 1 = K a.s. where Y 1 R d. (4.3 Y 2 Y 2 The process (V 0 is an MGOU process driven by he Lévy process ( X 3, Y 2 X 2 K 0 (4.4 V in R (m d (m d R m d. Furher, if (U, L is defined as in (2.9 and (3.5, and if V 0 is F 0 -measurable for a filraion F = (F 0 saisfying he usual assumpions such ha U and L are semimaringales wih respec o F (hence (V 0 solves he SDE (3.4 by Theorem 3.4, hen we have a.s. for each 0 ( ( U 1 U = 0 L 1 U 2 U 3 and L = L 2 wih U 1 R d d, L 1 R d, (4.5 where L 1 = U 1 K a.s. and (V 0 solves he SDE dv = du 3 V + d(l 2 + U 2 K, 0. (4.6 (b Conversely, if (4.2 and (4.3 hold for K = (k 1,..., k d R consan, hen he affine subspace H = {(k 1,..., k d, h d+1,..., h m, h d+1,..., h m R} of R m is invarian wih respec o he model (1.2 and for any saring random ( variable V 0 H he K MGOU process defined by (3.3 can be wrien as V = a.s., where (V 0 is an MGOU process driven by he Lévy process (4.4. V Remark 4.4. Observe ha if in he seing of Theorem 4.3 he invarian affine subspace H is no parallel o he axes, hen here exiss an orhogonal ransformaion marix O, such ha OH fulfills he assumpions of Theorem 4.3 for he ransformed MGOU process V = OV. The process (V 0 fulfills he random recurrence equaion V = A s,v s + B s, for 0 s where A s, = OA s, O and B s, = OB s, and hence by Theorem 3.1 i is an MGOU process driven by (OX O, OY 0. Thus he sudy of arbirary invarian affine subspaces reduces o he case reaed in Theorem

14 This observaion and Theorem 4.3 imply he following characerizaion of irreducibiliy of he model (1.2. Corollary 4.5. Suppose (X, Y 0 in R m m R m is a Lévy process such ha X fulfills (2.7. Then he auoregressive model (1.2 wih (A s,, B s, 0 s as defined in (3.1 is irreducible if and only if here exiss no pair (O, K of an orhogonal ransformaion O R m m and a consan K = (k 1,..., k d R d, 1 d m, such ha a.s. ( ( X OX O 1 = 0 X 1 and OY = K where X 1 R d d, 0. (4.7 X 2 X 3 Wih (U, L 0 as defined in (2.9 and (3.5, Equaion (4.7 is furher equivalen o ( ( U OU O 1 = 0 U 1 and OL = K a.s. wih U 1 R d d. (4.8 U 2 U 3 Y 2 L 2 5 Saionary Soluions of MGOU Processes In his secion we invesigae condiions for he exisence of sricly saionary soluions of mulivariae generalized Ornsein-Uhlenbeck processes. The proofs of he resuls are given in Secion 8. Given some exra informaion on he limi behaviour of E (X our firs heorem provides necessary and sufficien condiions for he exisence of saionary soluions of MGOU processes. Before we sae i we give he following lemma on sochasic exponenials which is ineresing in is own righ. Lemma 5.1. Le (X 0 be a Lévy process in R m m. Then for any 0 fixed we have ha E (X d = E (X. In paricular his implies P- lim E (X = 0 P- lim E (X = 0. (5.1 Since E (U = E (X, he condiion P- lim E (U = 0 appearing in Theorem 5.2(a below is equivalen o P- lim E (X = 0. Hence, Theorem 5.2 gives necessary and sufficien condiions for saionariy if eiher P- lim E (X = 0 or P- lim E (X = 0 and hus exends [3, Theorem 2.1]. Theorem 5.2. Suppose (V 0 is an MGOU process driven by he Lévy process (X, Y 0 in R m m R m such ha X saisfies (2.7. Le (U, L 0 be he Lévy process defined in (2.9 and (

15 (a Suppose lim E (U = 0 in probabiliy. Then a finie random variable V 0 can be chosen such ha (V 0 is sricly saionary if and only if he inegral E (Us dl s converges in disribuion for o a finie random variable. In his case, he disribuion of he sricly saionary process (V 0 is uniquely deermined and is obained by choosing V 0 independen of (X, Y 0 wih V 0 d = d- lim E (Us dl s. (b Suppose lim E (X = 0 in probabiliy. Then a finie random variable V 0 can be chosen such ha (V 0 is sricly saionary if and only if he inegral E (Xs dy s converges in probabiliy o a finie random variable as. In his case he sricly saionary soluion is unique and given by V = E (X E (Xs dy s a.s. for all 0. (, Observe ha he soluion obained in Theorem 5.2(a is causal and ha he one in (b is sricly non-causal. By adding he assumpion of irreducibiliy of he underlying model, as characerized in Corollary 4.5, he above heorem can be sharpened as follows. Theorem 5.3. Suppose (X, Y 0 is a Lévy process in R m m R m such ha X saisfies (2.7 and such ha he corresponding auoregressive model (1.2 wih (A s,, B s, 0 s as defined in (3.1 is irreducible. Le (V 0 be he MGOU process driven by (X, Y 0 and le (U, L 0 be he Lévy process defined in (2.9 and (3.5. (a A finie random variable V 0, independen of (X, Y 0, can be chosen such ha (V 0 is sricly saionary if and only if lim E (U = 0 in probabiliy and he inegral E (Us dl s converges in disribuion for o a finie random variable. (b A finie random variable V 0 can be chosen such ha (V 0 is sricly saionary and sricly non-causal if and only if lim E (X = 0 in probabiliy and he inegral E (Xs dy s converges in probabiliy as. Given ha he processes U and L have a finie log-momen Theorem 5.3 can be sharpenend o obain a necessary and sufficien condiion for he exisence of sricly saionary soluions of MGOU processes in erms of he driving Lévy process as saed in Theorem 5.4. To explain is condiions (iv and (v and relae i o he corresponding discree ime resuls, le be a fixed, submuliplicaive marix norm. Recall ha he op Lyapunov exponen of an R m m -valued i.i.d. sequence (C n n N wih E[log + C 1 ] is given by 1 γ := inf n N n E[log C 1 C n ]. (5.2 I is independen of he specific submuliplicaive marix norm used and i holds 1 γ = lim n n log C 1 1 C n = lim n n log C n C 1 a.s., (5.3 15

16 cf. Fursenberg and Kesen [11] and Bougerol and Picard [4]. In [4, Theorem 2.5] i is also shown ha if he discree ime model W n = C n W n + D n, n Z, is irreducible, where (C n, D n n Z is an i.i.d. R m m R m -valued sequence wih E[log + C 1 ] < and E[log + D 1 ] <, hen he discree model admis a sricly saionary causal soluion if and only if he op Lyapunov exponen of he sequence (C n n N is sricly negaive. Now if X and U are as in Theorem 5.4 (v, hen E[log + U 1 ] < implies E[log + E (U ] < for every > 0 as will be shown in Proposiion 8.4 below. Since for each h > 0 he sequence (A (nh,nh = E (U nh E (U (nh n N is i.i.d. and E (U nh = A (nh,nh A 0,h, i follows ha here is h > 0 such ha he op Lyapunov exponen of (A (nh,nh n N is sricly negaive if and only if here is 0 > 0 such ha E[log + E (U 0 ] <, which is equivalen o condiion (iv below by Lemma 5.1. Theorem 5.4. Suppose (X, Y 0 is a Lévy process in R m m R m such ha X saisfies (2.7 and ha he corresponding auoregressive model (1.2 wih (A s,, B s, 0 s as defined in (3.1 is irreducible. Le (V 0 be he MGOU process driven by (X, Y 0 and le (U, L 0 be he Lévy process defined in (2.9 and (3.5. Suppose ha E[log + U 1 ] < and E[log + L 1 ] <. Then he following are equivalen: (i A finie random variable V 0, independen of (X, Y 0, can be chosen such ha (V 0 is sricly saionary. (ii I holds lim E (U = 0 in probabiliy and he inegral E (Us dl s converges in disribuion for o a finie random variable. (iii I holds lim E (U = 0 a.s. and he inegral E (Us dl s converges a.s. for o a finie random variable. (iv There exiss 0 > 0 such ha E[log E (U 0 ] < 0. If addiionally U is a compound Poisson process wih jump heighs (S k k N, hen he above condiions (i o (iv are furher equivalen o (v The op Lyapunov exponen of he sequence (I + S k k N is sricly negaive. Remark 5.5. (a A similar resul as Theorem 5.4 also holds rue for sricly non-causal sricly saionary soluions of MGOU processes in he irreducible case. (b The proof of Theorem 5.4 given in Secion 8 shows ha he implicaions (iv = (iii = (ii = (i and (v = (iii = (ii = (i also hold wihou assuming irreducibiliy of he underlying model. Example 5.6. Consider he sae vecor process (V 0 of he COGARCH(q, m-volailiy process (S 0 as defined in Example 3.6 wih dv = du V + dl and (U, L 0 given by (3.12. Suppose ha m = 2. Then i follows from Corollary 4.5 by a sraighforward bu edious calculaion, using ha c 2 0, ha he corresponding auoregressive 16

17 model (1.2 is irreducible. In paricular, by Theorem 5.3(a, a sricly saionary (causal COGARCH(q, 2-volailiy sae vecor process exiss if and only if lim E (U = 0 in probabiliy and E (Us dl s = β 0<s E (Us e( M s 2 converges in disribuion o a finie random variable as. If in addiion log x ν x >1 M(dx <, where ν M denoes he Lévy measure of M, hen E log + U 1 < and E log + L 1 <, and by Theorem 5.4 he above condiions are equivalen o E log + E (U 0 < 0 for some 0 > 0. Tha he laer condiion is sufficien for a (causal sricly saionary sae vecor o exis was already observed in Remark 3.4(a of [5], bu having he irreducibiliy of he model we now also know ha i is necessary under he finie log-momen assumpion on ν M. Observe however ha he volailiy process (S 0 defined in (3.10 may be sricly saionary even wihou (V 0 being sricly saionary, since i is only a specific linear combinaion of (V 0 plus a consan. We shall no pursue he issue of sric saionariy of (S 0 furher. Also, we have no invesigaed if he auoregressive model (1.2 for he COGARCH(q, m volailiy process wih m 3 is always irreducible. In he case ha he underlying model is no irreducible, P- lim E (U = 0 is no necessary for he exisence of a causal sricly saionary soluion as shown in he following heorem. Theorem 5.7. Suppose (X, Y 0 is a Lévy process in R m m R m such ha X saisfies (2.7 and le (V 0 be he MGOU process driven by (X, Y 0 saisfying he auoregressive model (1.2 wih (A s,, B s, 0 s as defined in (3.1. Define (U, L 0 via (2.9 and (3.5. Then a finie random variable V 0 can be chosen such ha (V 0 is sricly saionary and causal if and only if here exiss a pair (O, K of an orhogonal ransformaion O R m m and a consan K = (k 1,..., k d, 0 d m such ha (4.7 and hence (4.8 hold and such ha P- lim E (U 3 = 0 and E (U 3 s d(l 2 s + U 2 sk converges in disribuion o a finie random variable as. If hese condiions are saisfied a sricly saionary soluion can be obained by choosing V 0 independen of (X, Y 0 wih he same disribuion as he disribuional limi as of ( K O. E (U 3 s d(l 2 s + U 2 sk If d = 0 in he above condiions hen L 2 s + U 2 sk has o be inerpreed as L 2 s, and if d = m hen U 3 is zero-dimensional and he convergence condiions regarding E (U 3 and E (U 3 s d(l 2 s + U 2 sk do no appear. Remark 5.8. Using argumens as in he proof of Theorem 5.3(b a similar resul as Theorem 5.7 for sricly noncausal sricly saionary soluions of MGOU processes can be obained, oo. 17

18 Remark 5.9. The resuls in Secions 3 and 5 remain valid if we rea an MGOU process (V 0 wih V R m l and drop he condiion of l = 1. As he value of l has no influence on he proofs we can simply replace he vecor valued processes (Y 0 and (L 0 by R m l - valued processes. Theorem 4.3 may be applied column-by-column or, alernaively, i is possible o inerpre he MGOU process (V 0 in R m l driven by (X, Y 0, X R m m, Y = (Y 1,..., Y l R m l as an MGOU process in R ml driven by he Lévy process X 0 Y 1...,. in R ml ml R ml. 0 X 6 Proofs for Secion 3 Y l Before proving Theorem 3.1 we give he following proposiion which esablishes in paricular he semimaringale propery of (A 0 and (B 0. Proposiion 6.1. Suppose (A s,, B s, 0 s is a process saisfying Assumpion 1 and such ha (A, B 0 is càdlàg. Le H be he naural augmened filraion of (A, B 0. Then (A 0 and (B 0 are H-semimaringales. Furher, he R m m R m R m m R m - valued process (U, L, X, Y 0 defined by U L X Y is an H-Lévy process. = 0 Log A = da s A s B da s A s B s Log A = A s da s A s d(a s B s, 0, (6.4 Proof. Observe ha (A 0 is a righ H-Lévy process by Assumpion 1 and Lemma 2.1 and hence an H-semimaringale by Proposiion 2.4. I follows ha (U, L, X 0 as given in (6.4 is well defined. By compuaions similar o hose in he proof of [12, Theorem 2.2] one can show ha for 0 s U U s L L s X X s = (s,] d(a s, u A s,u B s, (s,] d(a s, u A s,u B s,u (s,] A s,u d(a s, u, 0 s. (6.5 By Assumpion 1(b,c we observe ha (A s,s+u, B s,s+u d u 0 = (A 0,u, B 0,u u 0 and hus we obain from (6.5 ha (U, L, X has saionary incremens. By Assumpion 1(b, (U U s, L L s, X X s is independen from H s for 0 s, where H = (H 0. We also know ha (U 0, L 0, X 0 = 0 a.s., ha he pahs of (U, L, X are càdlàg since ha held rue for (A, B 0, and ha clearly (U, L, X is adaped o H. Hence (U, L, X 0 is an H-Lévy process. In paricular, L is an H-semimaringale, so ha by (6.4, (B 0 = (L + da s A s B s 0 18

19 is an H-semimaringale, oo. Consequenly Y as given in (6.4 is well defined. For 0 s we hen have from Assumpion 1(a ha Y Y s = A u d(a u B u (s,] = A s,u A s d(a s A s, A s, B s + A s A s, B s, u (s,] = A s,u d(a s, B s, u. (s,] I now follows in complee analogy o he reasoning given above ha (U, L, X, Y is an H-Lévy process. Proof of Theorem 3.1. Le (A s,, B s, 0 s saisfy Assumpion 1. By Proposiion 6.1, (A 0 and (B 0 are semimaringales wih respec o heir naural augmened filraion, and (X, Y defined by (3.2 is a Lévy process. Clearly, X saisfies (2.7, and for 0 s i holds A s, = A A s = (A A s Furhermore, A dy = d(a B from (3.2, so ha B = A A u dy u = E (X giving B s, = B A s, B s = E (X = E (X (s,] E (Xu dy u E (X E (Xu dy u. = E (X E (Xs. E (Xu dy u, E (Xs E (X s This is (3.1. The uniqueness of (X, Y is clear from (3.1. (0,s] E (Xu dy u For he converse, le (X, Y a Lévy process in R m m R m such ha X saisfies (2.7. Le F be he augmened naural filraion of (X, Y 0, hen (A s,, B s, 0 s as given in (3.1 is well defined wih respec o F and we know from Proposiion 2.4 ha E (X is a righ F-Lévy process in GL(R, m whose lef incremens are given by A s,. Thus we have ha for all 0 s u almos surely A s, = A u, A s,u holds. Also i follows direcly from he definiions of A s, and B s, ha B s, = A u, B s,u + B u, a.s. such ha Assumpion 1(a is fulfilled. For he common process (A s,, B s, 0 s observe ha for 0 s we have ( As, = E (X E (Xs B. s, E (X E (Xs (s,] E (X s E (Xu dy u 19

20 Since (A 0 is a righ F-Lévy process he common incremens ( E (X E (Xs, Y Y s s are independen of (X u, Y u 0 u s. Hence i follows ha {(A s,, B s,, a s b} and {(A s,, B s,, c s d} wih b c are independen. Similarly we conclude ha ( As, B s, d = which yields Assumpion 1(c. E (X s E (X0 E (X 0 E (X s E (X0 (0, s] E (Xu dy u = ( A0, s B 0, s The coninuiy in probabiliy a 0 of A = A 0, is clear, while for B = B 0, i follows from ha of A and Y and he coninuiy of he inegral. Proof of Theorem 3.4. (a I is easy o see ha (U, L as consruced in (2.9 and (3.5 is a Lévy process and ha U saisfies (3.6. Define A = A 0, and B = B 0, for 0 by he righ hand side of (3.1. Then V = A V 0 + B. By he definiion of U, we furher have ha da = du A. Hence, denoing L := B da u A u B u as in (6.4, we obain dv = da V 0 + db = du A V 0 + db = du (A V 0 + B + db du B = du V + db da A B = du V + dl. I remains o show ha L = L. Using he inegraion by pars formula (2.3 and (2.2, we obain L = B da s A s B s = E (X E (Xs dy s d( E (X s E (Xu dy u (0,s = E (X E (Xs dy s E (X E (Xs dy s ( ] E (Xu dy u E (Xu dy u + E (X s d (0,s] [ = E (X s E (Xs dy u + [ ] = Y + da s A s, Y (0, ] = Y + [U, Y ]. (0, ] + [ E (X, (0, ] d( E (X s E (X s, Y Tha L = L hen follows from he definiion of U in (2.9 since [ ] ( [U, Y ] = [X, Y ] + [[X, X] c, Y ] + (I + Xs I + X s, Y = [X, Y ] c + 0<s 0<s ( (I + Xs I Y s. ] 20

21 (b Is is clear ha (X, Y as defined in (3.7 is a Lévy process wih X saisfying (2.7. Observe ha he given definiion of (X 0 is equivalen o (2.9 and ha from he definiion of (Y 0 we deduce Y = L + [X, L] = L + ( X s L s + [X, L] c 0<s = L 0<s ( (I + Xs I (I + X s L s + [X, L] c, 0. Hence Y = (I + X L and [X, Y ] c = [X, L] c and we conclude ha Y = L ( (I + Xs I Y s + [X, Y ] c, 0, 0<s ( E ( which is equivalen o (3.5. Thus he MGOU process (X V 0 + E (Xs dy s 0 solves he SDE (3.4 by par (a, giving he claim by he uniqueness of he soluion o (3.4. Proof of Proposiion 3.5. Equaion (3.8 has been esablished when showing ha L = L in he proof of Theorem 3.4(a. Equaion (3.9 follows from he fac ha by (3.5, L = (I + X Y and [X, L] c = [X, Y ] c, so ha by he same calculaion as in he proof of Theorem 3.4(b, L + [X, L] = L ( (I + Xs I Y s + [X, Y ] c, 0. 0<s By (3.5 his implies ( Proofs for Secion 4 In his secion we give he proofs of he resuls of Secion 4. Proof of Proposiion 4.2. For ν N 0 le C ν := A 0,2 ν, D ν := B 0,2 ν and consider he random affine ransformaion f ν : R m R m, x C ν x + D ν, ν N 0. For 0 d < m, a d-dimensional affine subspace H of R m will be called an affine d-fla, and i is f ν -invarian if f ν (H = C ν H + D ν H a.s., which by Assumpion 1(c is equivalen o saying ha H is invarian for he discree ime model V nh = A (nh,nh V (nh + B (nh,nh, n N 0, as defined in (4.1 wih h = 2 ν. Since by (1.3 any subspace which is invarian for he model (4.1 for some h > 0 is also invarian for he model (4.1 for every h := kh wih k N, i is clear ha any f ν invarian affine d-fla is also f ν -invarian. Hence, denoing he se of all f ν -invarian affine d-flas by Hν, d 0 d < m, i follows ha Hν+1 d Hν d 21

22 for all ν N 0, 0 d < m, such ha H d := lim n Hν d = ν=0 Hd ν can be defined for 0 d < m. We furher denoe H ν := m d=0 H d ν for ν N 0 and H := m d=0 H d = H ν. The proof of he proposiion will be given in wo seps: firs i will be shown ha irreducibiliy of he coninuous ime model (1.2 implies H =, and in a second sep ha H = implies he exisence of some ν 0 N 0 such ha H ν0 =, i.e. ha he discree ime model (4.1 is irreducible for h = 2 ν 0. The firs sep will be shown by conradicion, i.e. we assume ha H, i.e. ha here exiss an affine subspace H R m of R m which is invarian under f ν for all ν N. Thus, as argued above, H is also invarian under he model (4.1 for all h = k2 ν, k N, ν N. I hen remains o show ha A 0, H + B 0, H holds for all > 0, i.e. ha H is invarian under (1.2. Bu his follows easily from he fac ha (A 0, 0 and (B 0, 0 have almos surely càdlàg pahs, so ha for every number > 0 we can find a sequence ( n n N of he form n = k n 2 νn converging from he righ o such ha almos surely, A 0, H + B 0, = lim n (A 0,n H + B 0,n H. Hence H is an invarian affine subspace of he coninuous ime model (1.2 giving he desired conradicion. I remains o show ha H = m d=0 Hd = implies he exisence of some ν 0 such ha H ν0 =. Since any affine 0-fla H of he form H = {x} is f ν -invarian if and only if (f ν I(x = 0 a.s., and since f n I is an affine linear mapping, is kernel is an affine linear subspace of R m, S ν say, and we have Hν 0 = {{x}, x S ν }. Since S ν+1 S ν, i follows ha here is ν 1 N such ha S ν1 +n = S ν1 for all n N 0. Hence, H = implies ha here is ν 1 N such ha Hν 0 = for all ν ν 1, and in he following we can concenrae on invarian affine d-flas wih 0 < d < m. Fix a family (O, K = {(O H, K H, H affine d-fla wih 0 < d < m} of pairs of an orhogonal ransformaion O H R m m and a consan K H R m such ha H := O H H K H = {0} m d R d. Then given ν N and some affine d-fla H we obain by easy compuaions ha H is invarian under f ν : x C ν x + D ν if and only if he subspace H is invarian under he mapping g ν : R m R m, x C ν x + D ν wih C ν = O H C ν O H and D ν = O H D ν + (O H C ν O H IK H. Using he special srucure of H his yields ha H is invarian under g ν if and only if i holds almos surely C (i,j ν = 0 and D (i ν ν=0 = 0 for all (i, j J H := {(i, j, 1 i d, d < j m}. This is by definiion of C ν and D ν equivalen o sae ha, almos surely, m m C(k,l ν (O O(j,l C(k,l ν = 0 and (7.1 m k=1 k,l=1 O (i,k H O (i,k H D(k ν + ( m m q=1 k,l=1 H (l,j = O (i,k H k,l=1 O(q,l H C(k,l ν O (i,k H H K (q H K(i H = 0 for all (i, j J H. (7.2 22

23 By inroducing he marices M H,i,j and N H,i in R m m via M (k,l H,i,j := O(i,k H O(j,l H and N (k,l H,i := m q=1 O (i,k H O(q,l H K(q H, k, l = 1,..., m, denoing he ih row of he marix O H by O (i, H and leing vec ( : Rm m R m2 be he vecorizaion operaor which sacks he colums of a given marix below one anoher, (7.1 and (7.2 urn ou o be equivalen o vec (M H,i,j vec (C ν vec (N H,i vec (C ν 0, D ν = 0 = (O (i, H, D ν (i, j J H, 0 K (i H almos surely, where, denoes he sandard scalar produc in R m2 +m+1. Now se R ν := span H H ν \H 0 ν vec (M H,i,j vec (N H,i 0, (O (i, H ; (i, j J H 0 K (i +m+1 Rm2, H where span denoes he linear span. Then by he above we have esablished ha R ν is orhogonal o (vec (C ν, Dν,, a.s., and ha, given an affine d-fla H, 0 < d < m, i is invarian under f ν if and only if all corresponding vecors (vec (M H,i,j, 0, 0 and (vec (N H,i, O (i, H, K(i H for (i, j J H are in R ν. Finally, as R ν is a vecor space and we have ha R ν+1 R ν we observe ha is limi for ν can only be empy (which is equivalen o m d=1 Hd = if here exiss some ν 2 N such ha for all ν ν 2 we have R ν =. Hence i holds H ν = for all ν ν 0 := max{ν 1, ν 2 } so ha he discree ime model (4.1 is irreducible for all ν ν 0 as had o be shown. Proof of Theorem 4.3. (a We sar by verifying (4.2 and (4.3. Since H is an invarian affine subspace we deduce from (1.2 ha for any 0 and all h d+1,..., h m R he equaion A (k 1,..., k d, h d+1,..., h m + B = (k 1,..., k d, g d+1,..., g m a.s. has o admi a soluion g d+1,..., g m R. This is equivalen o d j=1 d j=1 k j A (i,j + k j A (i,j + m j=d+1 m j=d+1 h j A (i,j + b i = k i, i = 1,..., d h j A (i,j + b i = g i, i = d + 1,..., m. Thus we can conclude ha A (i,j = 0 holds a.s. for i d, j > d. Observe by simple algebraic calculaions ha if wo marices M and N in R m m have a d (m d block of 23

24 zero enries in he upper righ corner, hen so do M and MN. More deailed we have for ( ( M1 0 N1 0 M = GL(R, m and N = R m m, M M 2 M 3 N 2 N 1, N 1 R d d, 3 ha M 1 and M 3 are non-singular and i holds ( ( M M 1 0 M = M 3 M 2 M 1 M and MN = 1 N M 2 N 1 + M 3 N 2 M 3 N 3 Now recall ha A = E (X and hus we know ha E (X and E (X a.s. admi a d (m d zero block for all 0. Hence i follows from (2.8 ha also X a.s. has such a zero block which is (4.2. Thus we deduce from (2.4 ha ( E 1 E (X =: 0 E 2 E 3 = ( I + 0 E1 s dx 1 s 0 0 E2 s dx 1 s + 0 E3 s dx 2 s I +, 0, (7.3 0 E3 s dx 3 s and observe in paricular ha E 1 = E (X 1 and E 3 = E (X 3 hold for 0. Insering he previous resuls in (3.3 yields for all 0 a.s. ( ( [( ( ( ] K (E V = = 1 0 K E 1 V (E 3 E 2 (E 1 (E 3 + s 0 Y 1 V 0 E 2 s E 3 d s s Y 2 s ( (E = 1 K (E 3 E 2 (E 1 K + (E 3 (7.4 V 0 ( (E E1 s dy 1 s (E 3 E 2 (E 1 0 E1 s dy 1 s + (E 3 [ 0 E3 s dy 2 s +. 0 E2 s dy 1 s] The equaion for he firs d enries of (7.4 is equivalen o K + E (X 1 s dy 1 s = E (X 1 K = K + E (X 1 s dx 1 sk a.s., 0, from where we deduce (4.3. From he equaion for he second m d enries of (7.4 we derive under use of (4.3, (2.4 and (7.3 ha V E (X 3 V 0 = E (X 3 = E (X 3 = E (X 3 = E (X 3 ( ( ( E 3 s dy 2 s + E 3 s dy 2 s + E 3 s dy 2 s + E 2 s dx 1 sk E 2 (E 1 (I + E 2 s dx 1 sk E 2 ( E 2 s dx 1 sk E (X 3 s d(y 2 s X 2 sk a.s., 0, E (X 1 E (X 1 K E 2 s dx 1 s + 0,] E 1 s dx 1 s K E 3 s dx 2 s K 24

25 such ha (4.4 is shown. Finally le (U, L 0 be he Lévy process defined in (2.9 and (3.5. By Theorem 3.4, (V 0 solves he SDE (3.4 wih respec o F. Observe ha by he same argumenaion as for X or alernaively by (2.9 we deduce ha for all 0 i holds U (i,j = 0 a.s. for i d, j > d. By insering U and L as given in (4.5 in he SDE (3.4 we obain L 1 = U 1 K in he firs and (4.6 in he second componen. This complees he proof. (b Insering (4.2 and (4.3 in (3.3 direcly gives he assumpion by calculaions similar as under (a. Proof of Corollary 4.5. By Remark 4.4, i only remains o show he equivalence of (4.7 and (4.8. For ha, observe ha E (U = E (X and hus E (OUO = E (OXO. Hence, as shown in he proof of Theorem 4.3, OXO has a d (m d block of zero enries in he upper righ corner, if and only if he same is rue for E (OXO, equivalenly for E (OXO, and hence equivalenly for OUO = Log ( E (OXO. I follows ha OX O is of he form as specified in (4.7 if and only if OU O is of he form specified in (4.8, and as seen in he proof of Theorem 4.3, i furher holds ha E (U 1 = E (X 1. To see he equivalence of he relaions regarding OY and OL, suppose firs ha Y saisfies (4.7. Then by (3.8, ( X OL = OY + [OUO 1, OY ] = K + Y 2 [( U 1 0 U 2 U 3, ( ] X 1 K Y 2, and he upper d componens on he righ hand side of his equaion are given by X 1 K + [U 1, X 1 K] = U 1 K, where he las equaion follows from (2.10 since E (U 1 = E (X 1. I follows ha L saisfies (4.8. Conversely, if (4.8 holds, hen i follows from (3.9 ha ( U OY = OL + [OXO 1, OL] = K + L 2 [( X 1 0 X 2 X 3, ( U 1 K L 2 ], and as above i follows from his equaion and (2.10 ha Y saisfies ( Proofs for Secion 5 In his secion we give he proofs for Secion 5 along wih a few resuls on mulivariae sochasic exponenials which will be needed bu are also ineresing in heir own righ. We sar by inroducing an approximaion of he sochasic exponenial which will be a useful ool. Namely, he following resul is due o Emery [8]. 25