Continuous time autoregressive moving average processes with random Lévy coefficients

Transcription

1 ALEA, La Am J Probab Mah Sa 4, (207) Coninuous ime auoregressive moving average processes wih random Lévy coefficiens Dirk-Philip Brandes Ulm Universiy Insiue of Mahemaical Finance, Helmholzsr 8, 8908 Ulm, Germany address: dirkbrandes@uni-ulmde URL: hp://wwwuni-ulmde/mawi/finmah/people/dirk-brandes/ Absrac We inroduce a coninuous ime auoregressive moving average process wih random Lévy coefficiens, ermed RC-CARMA(p,q) process, of order p and q < p via a subclass of mulivariae generalized Ornsein-Uhlenbeck processes Sufficien condiions for he exisence of a sricly saionary soluion and he exisence of momens are obained We furher invesigae second order saionariy properies, calculae he auocovariance funcion and specral densiy, and give sufficien condiions for heir exisence Inroducion Le q < p be non-negaiveinegersand L = (L ) R a Lévyprocess, ie a process wih saionary and independen incremens, càdlàg sample pahs and L 0 = 0 almos surely, which is coninuous in probabiliy A CARMA(p,q) process S = (S ) R driven by L is defined via S = b X, R, () wih X = (X ) R a C p -valued process which is a soluion o he sochasic differenial equaion (SDE) dx = AX d+edl, R, (2) where A =, e =, and b = a p a p a p 2 a b 0 b b p 2 b p Received by he ediors December 2h, 206; acceped February 8h, Mahemaics Subjec Classificaion 60G0, 60G5, 60H05, 60H0, 62M0 Key words and phrases CARMA process, coninuous ime auoregressive moving average process, mulivariae generalized Ornsein-Uhlenbeck process, Lévy process, coninuous ime, random coefficiens, saionariy, second order saionariy 29

2 220 Dirk-Philip Brandes wih a,,a p,b 0,,b p C such ha b q 0 and b j = 0 for j > q For p = he marix A is inerpreed as A = ( a ) I is well-known ha he soluion of (2) is unique for any X 0 and given by ) X = e (X A 0 + e As edl s, 0 (0,] Processes of his kind were firs considered for L being a Gaussian process by Doob (944) Brockwell (200b) gave he now commonly used definiion wih L being a Lévy process A CARMA process S = (S ) R as defined in () and (2) can be inerpreed as a soluion of he p h -order linear differenial equaion a(d)s = b(d)dl, where a(z) = z p +a z p + +a p, b(z) = b 0 +b z+ +b p z p and D denoes he differeniaion operaor In his sense, CARMA processes are a naural coninuous ime analog of discree ime ARMA processes Similar o ARMA processes, CARMA processes provide a racable bu rich class of sochasic processes Their possible auocovariance funcions h Cov(S,S +h ) are linear combinaions of (complex) exponenials and hus provide a wide variey of possible models when modeling empirical daa In discree ime, ARMA processes wih random coefficiens (RC-ARMA) have araced a lo of ineres recenly, in paricular, AR processes wih random coefficiens, see eg Nicholls and Quinn (982) They have applicaions as non-linear models for various processes, eg bilinear GARCH processes inroduced by Sori and Viale (2003) RC-ARMA processes also arise as a special case of condiional heeroscedasic ARMA (CHARMA) models proposed by Tsay (987) and are used for financial volailiy processes, see eg He and Teräsvira (999), o name jus a few As CARMA processes consiue he naural coninuous ime analog of ARMA processes, i is, herefore, naural o ask for CARMA processes wih random coefficiens The CARMA(, 0) process wih random (Lévy) coefficiens has already been sudied I is known as he generalized Ornsein-Uhlenbeck (GOU) process, which is obained as he soluion o he SDE dx = X dξ + dl, 0, where (ξ,l) = (ξ,l ) 0 is a bivariae Lévy process I has been shown by de Haan and Karandikar (989) ha GOU processes arise as he naural coninuous ime analog of he AR() process wih random iid coefficiens By choosing (ξ ) 0 = ( a ) 0, he GOU process reduces o he classical Lévy-driven Ornsein-Uhlenbeck process, which is a CAR(), ie CARMA(,0) process Boh he Ornsein-Uhlenbeck process as well as he generalized Ornsein-Uhlenbeck process have various applicaions in insurance and financial mahemaics, see eg Barndorff-Nielsen and Shephard (200) and Klüppelberg e al (2004) The aim of he presen paper is o inroduce CARMA processes wih random Lévy auoregressive coefficiens of higher orders, p, and o sudy saionariy andohernauralproperies Thedefiniionofourprocessisdoneinsuchawayha i includes he generalized Ornsein-Uhlenbeck process for order (, 0) as a special case and ha i reduces o he usual CARMA process when he auoregressive

3 Coninuous ime ARMA processes wih random Lévy coefficiens 22 Lévy coefficiens are chosen o be deerminisic Lévy processes, ie pure drif and henceforh linear funcions The paper is organized as follows In Secion 2 we give some preparaive resuls regarding mulivariae sochasic inegraion, he mulivariae sochasic exponenial, and mulivariae generalized Ornsein-Uhlenbeck processes In Secion 3 we define a CARMA process wih random coefficiens(rc-carma) and presen some basic properies as well as sufficien condiions for he exisence of a sricly saionary soluion Similar o Brockwell and Lindner (205) for CARMA processes, we furher show ha he RC-CARMA(p, 0) process saisfies an inegral-differenial equaion and examine is pah properies Secion 4 is concerned wih he exisence of momens, he auocovariance funcion, and he specral densiy, whereby i urns ou ha he laer wo have an ineresing connecion o hose of CARMA processes We end Secion 4 by invesigaing an RC-CARMA(2, ) process in more deail Conclusively, in Secion 5 we presen some simulaions 2 Preliminaries Throughou, we will always assume as given a complee probabiliy space (Ω,F,P) ogeher wih a filraion F = (F ) 0 By a filraion we mean a family of σ-algebras (F ) 0 ha is increasing, ie F s F for all s Our filraion saisfies, if no saed oherwise, he usual hypoheses, ie F 0 conains all P-null ses of F, and he filraion is righ-coninuous GL(R,m) denoes he general linear group of order m, ie he se of all m m inverible marices associaed wih he ordinary marix muliplicaion If A GL(R,m), we denoe wih A is ranspose and wih A is inverse For càdlàg processes X = (X ) 0 we denoe wih X and X := X X he lef-limi and he jump a ime, respecively A d-dimensional Lévy processes L = (L ) 0 can be idenified by is characerisic exponen (A L,γ L,Π L ) due o he Lévy-Khinchine formula, ie if µ denoes he disribuion of L, hen is characerisic funcion is, for z R d, given by [ µ(z) = exp ] 2 z,a Lz +i γ L,z + (e i z,x i z,x { x } (x))π L (dx) R d Here, A L is he Gaussian covariance marix which is in one dimension denoed by σ 2 L, Π L a measure on R d which saisfies Π L ({0}) = 0 and R d ( x 2 )Π L (dx) <, calledhelévy measure, andγ L R d aconsan Furher, x denoesheeuclidean norm of x For a deailed accoun of Lévy processes we refer o he book of Sao (203) Sochasic Inegraion A marix-valued sochasic process X = (X ) 0 is called an F-semimaringale or simply a semimaringale if every componen (X (i,j) ) 0 is a semimaringale wih respec o he filraion F For a semimaringale X R m n, and H R l m and G R n p wo locally bounded predicable processes, he R l p -valued sochasic inegral J = HdXG

4 222 Dirk-Philip Brandes is defined by is componens via n m J (i,j) = H (i,h) G (k,j) dx (h,k) k= h= I can easily be seen ha also in he mulivariae case he sochasic inegraion preserves he semimaringale propery as saed, for example, in he onedimensional case in Proer (2005) For wo semimaringales X R l m and Y R m n he R l n -valued quadraic covariaion [X,Y] is defined by is componens via [X,Y] (i,j) = m [X (i,k),y (k,j) ], (2) k= and similar is coninuous par [X,Y] c Then [X,Y] = [X,Y] c +X 0Y 0 + 0<s X s Y s, 0 (22) is rue also for marix-valued semimaringales Finally, as saed in Karandikar (99), for wo semimaringales X,Y R m m he inegraion by pars formula akes he form (XY) = 0+ X s dy s + The Mulivariae Sochasic Exponenial 0+ dx s Y s +[X,Y ], 0 Le X = (X ) 0 be a semimaringale in R m m wih X 0 = 0 Then is lef sochasic exponenial E (X) is defined as he unique R m m -valued, adaped, càdlàg soluion (Z ) 0 of he inegral equaion Z = I + Z s dx s, 0, (0,] where I R m m denoes he ideniy marix The righ sochasic exponenial of X, denoed as E (X), is defined as he unique R m m -valued, adaped, càdlàg soluion (Z ) 0 of he inegral equaion Z = I + dx s Z s, 0 (0,] I can be shown ha boh he lef and he righ sochasic exponenial are semimaringales and ha for is ranspose i holds E (X) = E (X ) As observed by Karandikar (99), he righ and he lef sochasic exponenials of a semimaringale X are inverible a all imes 0 if and only if de(i + X ) 0 0 (23) We also need he following resul of Karandikar (99) Proposiion 2 (Inverse of he Sochasic Exponenial) Le X = (X ) 0 be a semimaringale wih X 0 = 0 such ha (23) holds Define

5 Coninuous ime ARMA processes wih random Lévy coefficiens 223 he semimaringale U := X +[X,X] c + Then and Furher, 0<s and X can be represened by X = U +[U,U] c + ( (I + Xs ) I + X s ), 0 (24) [ ] E E (X) = (U ) = E (U) 0, (25) U = X [X,U] 0 (26) de(i + U ) 0 0, 0<s ( (I + Us ) I + U s ), 0 (27) Proof: For(25) and(26) see Karandikar(99), Theorem For he remaining asserions, observe ha U = (I+ X ) I from (24), so ha de(i+ U ) 0 for all 0 Furher, from (24) we obain [U,U] c = [X,X]c Insering his, U = (I + X ) I, and he form of U from (24) ino he righ-hand side of (27) gives X so ha (27) is rue Mulivariae Generalized Ornsein-Uhlenbeck processes We give a shor overview of resuls regarding mulivariae generalized Ornsein- Uhlenbeck (MGOU) processes which are used hroughou MGOU processes were inroduced by Behme and Lindner (202) and furher invesigaed in Behme (202) Definiion 22 Le (X,Y) = (X,Y ) 0 be a Lévy process in R m m R m such ha X saisfies (23), and le V 0 be a random variable in R m Then he R m -valued process V = (V ) 0 given by ( V = E (X) V 0 + (0,] E (X)s dy s ), 0, is called a mulivariae generalized Ornsein-Uhlenbeck (MGOU) process driven by (X,Y) The underlying filraion F = (F ) 0 is such ha i saisfies he usual hypoheses and such ha (X,Y) is a semimaringale The MGOU processwill be calledcausal ornon-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 Remark 23 I follows from Behme and Lindner (202), Theorem 34, ha an MGOU process V = (V ) 0 wih an F 0 -measurable V 0 solves he SDE dv = du V + dz, 0, (28) where U = (U ) 0 is anoher R m m -valued Lévy process defined by (24) so ha E (X) = E (U), and Z = (Z ) 0 is a Lévy process in R m given by Z = Y + ( (I + Xs ) I ) Y s [X,Y] c, 0 0<s

6 224 Dirk-Philip Brandes Wih hese U and Z he MGOU process can also be wrien as ) V = E (U) (V 0 + E (U) s dy s, 0 (29) (0,] Conversely, if (U,Z) = (U,Z ) 0 is a Lévy process in R m m R m such ha i holds de(i + U ) 0 for all 0, hen for every F 0 -measurable random vecor V 0 he soluion o (28) is an MGOU process driven by (X,Y), where X is given by (27) and Y = Z +[X,Z] Convenion 24 Observe ha an MGOU process and similarly he process given by (29) is well-defined for any saring random vecor V 0, regardless if i is F 0 - measurable or no We shall hence speak of (29) as a soluion o (28), regardless if V 0 is F 0 -measurable or no Observe ha if V 0 is chosen o be independen of (U,Z) or equivalenly (X,Y), hen he naural augmened filraion of (U,Z) may be enlarged by σ(v 0 ) such ha (U,Z) sill remains a semimaringale, see Proer (2005) Theorem VI2 Wih his enlarged filraion, V 0 is measurable To invesigae he sric saionariy propery of RC-CARMA processes laer, we inroduce he propery of irreducibiliy of a class of MGOU processes as i has been done in Secion 4 of Behme and Lindner (202) Definiion25 Supposeha(X,Y) = (X,Y ) 0 isalévyprocessinr m m R m such ha X saisfies (23) Then an affine subspace H of R m is called invarian for he class of MGOU processes V = (V ) 0 driven by (X,Y) if for all x H he choice V 0 = x implies V H as for all 0 If here exiss no proper affine subspace H such ha, for all x H, V 0 = x implies V H as for all 0, we call he class of MGOU processes irreducible Irreducibiliy is hus a propery of he considered model By abuse of language, we will call an MGOU process irreducible if he corresponding class saisfies Definiion 25 3 The RC-CARMA process Le p N and C = (C ) 0 = (M (),,M (p),l ) 0 be a Lévy process in R p+ wih Π M ()({}) = 0 Le b 0,,b p R Le U = (U ) 0 be R p p -valued defined by b U :=, b :=, (3) M (p ) M (p 2) M () M (p) b b p 2 b p e := [0,,0,] he p h uni vecor, and q := max{j {0,,p }: b j 0} Then we call any process R = (R ) 0 which saisfies where V = (V ) 0 is a soluion o he SDE R = b V, 0, (32) dv = du V +edl, 0, (33)

7 Coninuous ime ARMA processes wih random Lévy coefficiens 225 an RC-CARMA(p,q) process, ie a CARMA process wih random Lévy coefficiens We speak of C and b as he parameers of he RC-CARMA process As will be seen in Proposiion 34 below, he assumpion Π M ()({}) = 0 implies de(i + U ) 0 for all 0, so ha V is an MGOU process as in (28) and, as in Convenion 24, by a soluion of (33) we mean a process of he form (29) wih saring random variable V 0 no necessarily F 0 -measurable We shall call he process V a sae vecor process of he RC-CARMA process R Observe ha we ge a classical CARMA(p,q) process (S ) 0 = (b V ) 0, alhough on he posiive real line, by choosing (M (),,M (p) ) = (a,,a p ) wih a,,a p R Furher, we recognize ha here is less sense in choosing he coefficiens of he moving average side o be random since hey are jus defining he weighs of he componens of V o form S Recall ha a CARMA(p,q) process S = (S ) R saisfies he formal p h -order linear differenial equaion a(d)s = b(d)dl, (34) where a(z) = z p +a z p + +a p, b(z) = b 0 +b z+ +b p z p, and D denoes he differeniaion wih respec o When we consider an RC-CARMA(p,0) process (R ) 0 = (b 0 V ) 0 for (V ) 0 = (V,,V p ) 0 solving (33), we formally find ha where a M (D)R = a M (D)b V = a M (D)b 0 V = b 0 DL = b(d)dl, (35) a M (z) = z p + dm() d z p + + dm(p) z 0 d Observe ha dm(i) d is no defined in a rigorous way bu jus an inuiive way of wriing Thus, i is possible o inerpre also he RC-CARMA(p,0) process as a soluion o a formal p h -order linear differenial equaion wih random coefficiens To jusify (35), look a he firs p componens of V dv i = V i+ d dv i d Formal division by d yields for he p h componen (36) = V i+ D i V = V i+, i =,,p (36) dv p = V dm(p) V p dm () + dl dv p d = V dm (p) d DV p = V dm (p) d DL = D p V + dm() DL = a M (D)V V p dm () d D p V d D p V + dl d dm () d +DL dm(p) + + V d Brockwell and Lindner (205) gave a rigorous inerpreaion of (34) by showing ha a CARMA(p,q) process (S = b X ) R driven by a Lévy process L saisfies he inegral equaion a(d)j p S = b(d)j p (L )+a(d)j p (b e A X 0 ), R,

8 226 Dirk-Philip Brandes where a(z) and b(z) are as before, and J denoes he inegraion operaor which associaes wih any càdlàg funcion f = (f ) R : R C, f, he funcion J(f) defined by J(f) := 0 f s ds Similarly, we give a rigorous inerpreaion of (35) as an inegral-differenial equaion and show ha he RC-CARMA(p, 0) process solves his equaion, hence making he formal deviaion of (35) above horoughly We call a funcion g: [0, ) R differeniable wih càdlàg derivaive Dg, if g is coninuous and here exiss a càdlàg funcion Dg such ha g is a every poin R righ- and lef-differeniable wih righ-derivaive Dg and lef-derivaive Dg = lim ε 0,ε 0 Dg ε, respecively In oher words, g is absoluely coninuous and has càdlàg densiy Dg (see he discussion in Brockwell and Lindner, 205 a he beginning of Secion 2) We call a funcion g: [0, ) R p-imes coninuously differeniable wih càdlàg derivaive D p g, if g is p -imes differeniable in he usual sense and he (p ) s derivaive D (p ) g is differeniable wih càdlàg derivaive D p g = D(D p g) as defined above Theorem 3 Le C = (C ) 0 = (M (),,M (p),l ) 0 be a Lévy process in R p+ and an F-semimaringale wih Π M ()({}) = 0 Le b = [b 0,,b p ] R p wih b 0 0 and b = = b p = 0, and consider he RC-CARMA(p,0) process R = (R ) 0 defined by (32) and (33), where V = (V ) 0 is he sae vecor process (wih V 0 no necessarily F 0 -measurable) Denoe by D p he se of all F- adaped, R-valued processes G = (G ) 0 which are p imes differeniable wih càdlàg derivaive D p G (a) Define W = (W ) 0 by W := R b E (U) V 0, 0 Then W D p, i is an RC-CARMA process wih parameers C, b, and iniial sae vecor 0, and i saisfies he inegral-differenial equaion ( ) D p W + D i W s dm s (p i+) = b 0 L, 0 (37) i= (0,] If V 0 is addiionally F 0 -measurable, hen also R D p and here exiss an F 0 - measurable random variable Z 0 such ha ( ) D p R + D i R s dm s (p i+) = b 0 L +Z 0, 0 (38) i= (0,] (b) Conversely, if R = ( R ) 0 D p saisfies ( ) D p R + D i Rs dm s (p i+) = b 0 L +Z 0 (39) i= (0,] for some F 0 -measurable Z 0, hen R is an RC-CARMA process wih parameers C, b, and sae vecor process Ṽ = (Ṽ) 0 := (b 0 ( R,D R,,D p R )) 0 Especially, Ṽ0 is F 0 -measurable

9 Coninuous ime ARMA processes wih random Lévy coefficiens 227 Proof: (a) As already observed (and o be shown in Proposiion 34 (a) below), he condiion Π M ()({}) = 0 implies ha V is an MGOU process So ) V = E (U) (V 0 + E (U) s dy s (0,] for some Lévy process Y as specified in Remark 23 Hence, (V E (U) V 0 ) 0 is adaped and consequenly so is (R b E (U) V 0 ) 0, and i is obviously an RC- CARMA process wih iniial sae vecor 0 Hence, for he proof of (37) i suffices o assume ha V 0 = 0 Denoe V = (V ) 0 = (V,,V p ) 0 By (36), we have D i V = V i+, i =,,p Hence, V is p -imes differeniable wih (p ) s càdlàg derivaive V p By he defining SDE of he RC-CARMA process (33) and he form of he marix U = (U ) 0 given in (3), we also have and, since V 0 = 0, ha V p = V p 0 i= L = D p V + (0,] i= V i s dm (p i+) s +L, (30) (0,] D i V s dm (p i+) s (3) Muliplying (3) by b 0 gives (37) when V 0 = 0 and hence (37) in general For (38) le K = (K,,Kp ) := E (U) V 0 WhenweconsiderheSDEwhichdefinesherighsochasicexponenial d E (U) = du E (U), we obain and, due o he form of he process U, dk i = Ki+ d dki d dk = du K wih K 0 = V 0, (32) = K i+ Furher, from he las componen of (32) K p = Dp K = V p 0 Ks dmp s Hence, (0,] = V p 0 D p K + i= i= (0,] (0,] D i K = Ki+, i =,,p (0,] D i K s dm (p i+) s D i K s dm(p i+) s = V p 0, K p s dm s where V p 0 is F 0-measurable such ha we obain (38) via R = W +b K, where W saisfies (37)

10 228 Dirk-Philip Brandes (b) For he converse, le R D p saisfy (39), and denoe Ṽ = (Ṽ,,Ṽ p ) = 0 ( R,D R,,D p R )), 0 By he fundamenal heorem of calculus (b and from (39) we obain Ṽ i = Ṽ i 0 + Ṽ p = b 0 Z 0 = b 0 Z 0 (0,] i= i= Ṽ i+ s ds, i =,,p, (33) (0,] (0,] Bu (33) and (34) mean ha Ṽ p Ṽ = Ṽ0 + 0 = b D i Ṽs dm(p i+) s +L Ṽ i s dm(p i+) s +L, 0 (34) (0,] 0 Z 0 and ha Ṽ = (Ṽ) 0 saisfies du s Ṽ s +el Since obviously b Ṽ = b 0 b0 R = R, i follows ha R is an RC-CARMA(p,0) process wih parameers C, b, and sae vecor process Ṽ Remark 32 Differeniaing (38) formally gives D p R + D i R s DM s (p i+) = b 0 DL, i= hence (35), and he RC-CARMA(p, 0) process can be inerpreed as a soluion o a formal p h -order linear differenial equaion wih random coefficiens To obain a similar equaion and hence inerpreaion for RC-CARMA(p, q) processes wih q > 0 seems no so easy since i is in general no possible o inerchange he sochasic inegraion wih he differeniaion operaor D Remark 33 Similar as in case of CARMA(p,q) processes, we easily see for an RC-CARMA(p,q) process R = (R ) 0 wih q < p, b q 0, and b j = 0 for j > q wih R = b V = b 0 V + +b q V q+ ha, by (36), R is (p q )-imes differeniable wih (p q ) s càdlàgderivaive D p q R = b 0 D p q V + +b q D p q V q+ = b 0 V p q + +b q V p The following proposiion shows ha he sae vecor process V = (V ) 0 is an MGOU process and gives is specific driving Lévy process (X,Y) Furher, V is an irreducible MGOU process if we assume ha U is independen of L and L is no deerminisic Proposiion 34 Le C = (C ) 0 = (M (),,M (p),l ) 0 be an R p+ -valued Lévy process and an F-semimaringale Le Π M () denoe he Lévy measure of M () and le U be defined as in (3) (a) Then E (U) GL(R,p) for all 0 if and only if Π M ()({}) = 0 In his case, for any saring random vecor V 0 he soluion o dv = du V +edl

11 Coninuous ime ARMA processes wih random Lévy coefficiens 229 is an MGOU process driven by (X,Y), ie V = (V ) 0 akes he form ( ) ) V = E (X) V 0 + E (X)s dy s = E (U) (V 0 + E (U) s dy s (0,] (0,] for 0 Here, (X,Y) = (X,Y ) 0 is a Lévy process defined by X = N (p) N (p ) N (p 2) N () wih N (i) = M (i) +σ M (),M (i) + M s (i) M s () 0<s M s (), i =,,p, where σ M (),M (i) denoes he Gaussian covariance of M() and M (i) X saisfies de(i + X ) 0 for all 0, and Y = eỹ wih Ỹ = L +σ M (),L + 0<s M () s L s M () s (b) Denoe M = (M ) 0 = (M (),,M (p) ) 0 Assume ha M is independen of L and L no deerminisic Then he MGOU process V obained in (a) is irreducible Proof: (a) Since C is a Lévy process and a semimaringale wih respec o F, i is clear ha also U as defined in (3) is a semimaringale and herefore also E (U) is a semimaringale wih respec o F Since E (U) is non-singular for all 0 if and only if de(i + U ) 0 for all 0, we calculae de(i + U ) = de M (p) M (p ) M (2) M () = ( M () ) (35) which shows ha E (U) is non-singular if and only if M () for all 0 Since M () is a Lévy process, he laer is equivalen o Π M ()({}) = 0 By Remark 23, V is an MGOU process driven by (X,Y), where X is given by (27) and saisfies de(i + X ) 0 for all 0, and Y = el +[X,eL] From (2) we obain for he componens of [U,U] c due o he form of U in (3) { ([U,U] c )(i,j) = [U (i,k),u (k,j) ] c 0, i =,,p, = (p j+), i = p k= σ M (),M

12 230 Dirk-Philip Brandes The form of I + U is implicily given in (35) such ha (I + U ) = M (p) M () M (p ) M () M (2) M () M () which is well-defined since Π M ()({}) = 0 Summing up he erms according o (27) leads o he saed form of he processes (N (),,N (p) ) 0 and X For Y = el +[X,eL] we obain wih (2) componenwise { [X,eL] (i) = [X (i,k),(el) (k) ] = [X (i,p) 0, i =,,p,,l] = [N (),L], i = p Since k= N () = M() M () and [N (),L] c = [M(),L] c = σ M (),L we ge he saed form of Y by (22) (b) Suppose ha V = (V ) 0 = (V,,V p ) 0 is no irreducible Hence, here exiss an invarian affine subspace H wih dimh {0,,p } Then for all x H i holds P(V H V 0 = x) = for all 0 Since V is càdlàg, we obain P(V H 0 V 0 = x) = Furher, if H H wih dimh = p, hen, P(V H 0 V 0 = x) = x H (36) We shall show ha P(V H 0 V 0 = x) < for all x R p, hence conradicing (36) So assume wlog ha dimh = p Since hen H is a hyperplane, here exiss λ R p wih λ 0 and a R such ha H = {y R p : λ y = a} Le V 0 = x H Le i,,i k {,,p} wih i n i m for all n m, λ in 0 for all n {,,k} and λ j = 0 for j {,,p}\{i,,i k } Wlog assume i < i 2 < < i k By he exisence of an invarian affine subspace, his yields λ i V i + +λ ik V i k = a This is, since D in i V i = V in, n =,,k, equivalen o λ i V i +λ i2 D i2 i V i + +λ ik D i k i V i = a (37) Bu (37) is an inhomogeneousordinary linear differenial equaion of order i k i Hence, V i isasmoohdeerminisicfuncioninandsoarev j forallj {,,p} since V j = D j V When we consider he las componen of V, we obain by (30) wih V p 0 = xp L = V p + Vs k dm(p k+) s x p (38) k= (0,] BuunderheassumpionhaM andlareindependenandlisnodeerminisic, (38) canno hold (observe ha V is deerminisic by (37) when V 0 = x) This gives he waned conradicion and herefore ha V is irreducible Remark 35 We can wrie (38) using parial inegraion as L = V p + V k M (p k+) M (p k+) s dvs k x p, k= k= (0,]

13 Coninuous ime ARMA processes wih random Lévy coefficiens 23 hence L is a funcional of M I would herefore be enough assuming ha L is no measurable wih respec o he filraion generaed by M o ensure irreducibiliy of he sae vecor process V Remark 36 When he componens M (),,M (p),l of he Lévy process C in Proposiion 34 (a) are addiionally independen, hen he componens do no jump ogeher almos surely and he Gaussian covariances vanish for differen componens so ha he formulas for N (),,N (p) simplify o N () = M () +σ 2 + M () 0<s ( M () s ) 2 M () s and N (i) = M (i) as for i = 2,,p Furher, Y = el and X are independen in ha case (he laer is already rue when jus L is independen of (M (),,M (p) )) Recall ha a process is sricly saionary if is finie-dimensional disribuions are shif-invarian As in he case of he MGOU process, we can find sufficien condiions for he exisence of a sricly saionary soluion of he RC-CARMA SDE (33) and herefore a sricly saionary RC-CARMA process Assume hroughou ha he used norm on R p p is submuliplicaive Theorem 37 Le C = (C ) 0 = (M (),,M (p),l ) 0 be a Lévy process in R p+ wih Π M ()({}) = 0 and a semimaringale wih respec o he given filraion F Le b := [b 0,,b p ] R p, q := max{j {0,,p }: b j 0} and U = (U ) 0 be given as in (3) Assume ha E [ log + U ] < and E [ log + L ] < (a) Suppose here exiss a 0 > 0 such ha [ ] E log E < 0 (39) (U)0 Then E (U) converges as o 0 as, and he inegral (0,] E (U)s edl s converges as o a finie random vecor as, denoed by 0 E (U)s edl s Furher, (33) admis a sricly saionary soluion which is causal and unique in disribuion, and is achieved by choosing V 0 o be independen of C and such ha d V 0 = 0 E (U)s edl s (b) Conversely, if V 0 can be chosen independen of C such ha V is sricly saionary, M = (M (),,M (p) ) 0 is independen of L and L no deerminisic, hen here exiss a 0 > 0 such ha (39) holds In boh cases wih his choice of V 0, he RC-CARMA process given by R = b V, 0, is sricly saionary, oo Proof: (a) The asserions regarding V follow from Theorem 54 wih Remark 55 (b) and Theorem 52 (a) in Behme and Lindner (202) (b) Under he assumpions made, V is irreducible by Proposiion 34 (b) Therefore Theorem 54 of Behme and Lindner (202) applies Tha R is sricly saionary if V is, is obvious 4 Exisence of momens and second order properies In his secion, we calculae he auocovariance funcion (ACVF) of an RC- CARMA process and give a connecion o he auocovariance funcion of a specific

14 232 Dirk-Philip Brandes CARMA process obained by choosing A = E[U ] Furher, we give sufficien condiions for he exisence of he ACVF and he specral densiy We end his secion wih an exemplary invesigaion of he RC-CARMA(2, ) process Bu we sar wih a resul which guaranees he exisence of higher momens Assume ha he used norm on R p p is submuliplicaive Proposiion 4 Le R = (R ) 0 be an RC-CARMA(p,q) process wih parameers C = (C ) 0 = (M (),,M (p),l ) 0, b and sricly saionary sae vecor process V = (V ) 0 wih V 0 independen of C and C a semimaringale wih respec o he given filraion F Assume ha for κ > 0 we have for some 0 > 0 E C max{κ,} < and E κ E < (4) (U)0 Then E R 0 κ <, and if (4) holds for κ =, E[R 0 ] = b 0 E[L ] E[M (p) ] Remark 42 The assumpion (4) in he previous proposiion acually already impliesheexisenceofasriclysaionarysaevecorprocessv = (V ) 0, hais unique in disribuion, since E C max{κ,} < obviously implies E[log + U ] < and E[log + L ] <, and by Jensen s inequaliy and (4) we furher have [ ] κe log E loge κ (U)0 E < 0 (U)0 Then Theorem 37 applies Proof of Proposiion 4: By Remark 42 he sricly saionary sae vecor process V is unique in disribuion By Proposiion 33 in Behme (202) we hen have E V 0 κ < and hence E R 0 κ < Now le κ = Again from Behme (202), Proposiion 33, we know ha E[U ] is inverible and E[V 0 ] = E[U ] ee[l ] Observe ha E[U ] is a companion marix and i is well-known ha he inverse of his is of he form E[M(p ) ] E[M(p 2) E[M (p) ] E[M() ] E[M (p) ] ] E[M (p) ] E[M (p) ] such ha E[U ] = E[V 0 ] = e E[L ] E[M (p) ], where e denoes he firs uni vecor in R p, and E[R 0 ] = E[b V 0 ] = b e E[L ] E[M (p) ] = b 0 E[L ] (42) E[M (p) ]

15 Coninuous ime ARMA processes wih random Lévy coefficiens 233 The following proposiion gives sufficien condiions for he exisence of he auocovariance funcion of an RC-CARMA process and saes is form We denoe wih he Kronecker produc and by vec he vecorizing operaor which maps a marix H form R p m ino R pm sacking is columns one under anoher vec (H) means he inverse operaion and yields H Proposiion 43 Le R = (R ) 0 be an RC-CARMA(p,q) process wih parameers C = (C ) 0, b and sae vecor process V = (V ) 0 wih V 0 independen of C and C a semimaringale wih respec o he given filraion F Suppose i holds E C 2 < and E V s 2 <, hen, for 0, we have Cov(R +h,r ) = b e he[u] Cov(V )b h 0, where Cov(V ) = E[V V ] E[V ]E[V ] denoes he covariance marix of V In paricular, if V is sricly saionary, (4) holds for κ = 2 and we denoe D = E[U ] I +I E[U ]+E[U U ] E[U ] E[U ], (43) hen all eigenvalues of D have sricly negaive real pars and he marix F = s 0 is finie Furher, if E[L ] = 0, we obain 0 e ud (e (s u)(e[u] I) +e (s u)(i E[U]) )duds Cov(R +h,r ) = b e he[u] vec ( D e p 2)E(L 2 )b, where e p 2 denoes he (p 2 ) h -uni vecor in R p2, and if M = (M (),,M (p) ) 0 and L are independen, we obain Cov(R +h,r ) = b e he[u] (vec ( D e p 2Var(L )+Fe p 2(E[L ]) 2) e e ( E[L ] E[M (p) ] ) 2 ) Proof: This follows immediaely from Proposiion 34 and he subsequen remark in Behme (202), by observing ha vec ((E[U ] E[U ]) vec(e[el ]E[eL ] )) = E[U ] E[eL ]E[eL ] (E[U ] ) ( ) 2 = e e E[L ] E[M (p) ] is obained by (42), and he properies of he vecorizing and he Kronecker produc operaions The following is Remark 35 (a) in Behme (202) for our purposes Remark 44 Le C = (C ) 0 = (M (),,M (p),l ) 0 be a Lévy process in R p+ wih E C 2 < saisfying Π M ()({}) = 0 and U = (U ) 0 a Lévy process in R p p of he form (3) Le D be as in (43) Then E 2 E < for some (U)0 0 > 0 if and only if all eigenvalues of D have sricly negaive real pars b

16 234 Dirk-Philip Brandes Therefore, ha condiion (4) holds for κ = 2 can be replaced by E C 2 < and all eigenvalues of D have sricly negaive real pars (44) Le R = (R ) 0 be an RC-CARMA process wih parameers b and C = (M (),,M (p),l) If E M (),,E M (p) <, we can associae o R and each vecor X 0 a CARMA process S = (S ) 0, given by () and (2) wih A = E[U ] Each of hese processes, ie when X 0 varies over all random variables, will be called a CARMA process associaed wih he given RC-CARMA process wih sae vecor process (X ) 0 I is hen ineresing o compare he auocovariance funcion of R wih ha of S provided boh are sricly saionary wih finie variance We denoe wih he Kronecker sum, ie A A = A I +I A Theorem 45 Le R = (R ) 0 be an RC-CARMA(p,q) process wih parameers C = (C ) 0, b and sricly saionary sae vecor process V = (V ) 0 wih V 0 independen of C, and C a semimaringale wih respec o he given filraion F Assume ha (4) holds for κ = 2, ha E[L ] = 0 and denoe D := E[U ] E[U ] Then E[U ] and D have only eigenvalues wih sricly negaive real pars Furher, X 0 can be chosen independen of C and unique in disribuion such ha he sae vecor process (X ) 0 of he associaed CARMA process S = (S ) 0 becomes sricly saionary wih finie variance Is auocovariance funcion can be expressed for all 0 as Cov(S +h,s ) = b e he[u] vec ( D e p 2)E(L 2 )b h 0 (45) Then he auocovariance funcion of S and R differ only by a muliplicaive consan More precisely, where Cov(R +h,r ) = Cov(S +h,s ) RC,h 0, (46) RC = +e p 2B D e p 2 (47) wih B = E[U U ] E[U ] E[U ] Furhermore, if Var(R 0 ) > 0, he auocorrelaion funcions of boh R and S agree, ie Corr[S +h,s ] = Corr[R +h,r ],h 0 (48) Proof: Tha E[U ] has only eigenvalues wih sricly negaive real pars follows since (4), by Jensen s inequaliy, implies ( E[ E (U) 0 ] E ) E (U)0 E 2 /2 E < (U)0 Hence, since E[ E (U) 0 ] = e 0E[U] by Proposiion 3 in Behme (202) we obain by he submuliplicaiviy of he norm ( e n0e[u] e 0E[U ] n 2) n/2 E E (U)0 0, n, so ha all eigenvalues of E[U ] have sricly negaive real pars (eg Proposiion 82 in Bernsein, 2009) Tha hen also D has only eigenvalues wih sricly negaive real pars follows by Fac 7 of Bernsein (2009) Tha X admis a sricly saionary soluion which is unique in disribuion wih finie variance and X 0 independen of C under he given condiions, is well-known

17 Coninuous ime ARMA processes wih random Lévy coefficiens 235 (eg Brockwell, 200a) or follows alernaively from Remark 42 (45) follows from Proposiion 43 (48) is clearly rue as long as (46) holds and Var(R 0 ) 0 To show ha (46) is indeed rue, we recognize firs ha he covariance of he CARMA process S and he RC-CARMA process R differ only in he marices D as defined in (43) and D So, i is enough o show ha x RC = x where x := D e p 2 and x := D e p 2 and RC is defined by (47) Since boh D and D, under he assumpions made, are inverible, x and x are well-defined Furher, we have ha D = D +E[U U ] E[U ] E[U ] =: D+B, where he marix B = (b i,j ) i,j=,,p 2 does only have values differen from zero in he las row The laer can be seen due o he form of he marix U by 0 p E[U ] 0 p 0 p 0 p 0 p E[U ] 0 p E[U U ]=, 0 p 0 p 0 p E[U ] E[M (p) U ] E[M (p ) U ] E[M (p 2) U ] E[M () U ] and E[U ] E[U ] = 0 p E[U ] 0 p 0 p 0 p 0 p E[U ] 0 p 0 p 0 p 0 p E[U ] E[M (p) ]E[U ] E[M (p ) ]E[U ] E[M (p 2) ]E[U ] E[M () ]E[U ] Then e p 2 = D x = (D B) x = D x b p 2,i x i e p 2 such ha p 2 D x = + b p 2,i x i e p 2 = (+(e p 2B) D e p 2)e p 2 = RC e p 2 i= p 2 i= Hence, x = RC D e p 2 = RC x The following wo proposiions give handy sufficien condiions for (44) and herefore also for he exisence of a sricly saionary soluion by Remark 42 Proposiion 46 Suppose ha E C 2 < and ha E[U ] has only eigenvalues wih sricly negaive real pars Denoe D := E[U ] E[U ], and B := E[U U ] E[U ] E[U ] = (b i,j ) i,j=,,p 2 wih b i,j = 0 for all i p 2 and all j =,,p 2 and b p2,p(k )+j = Cov(M (p k+),m (p j+) ) for k,j =,,p Then he minimal

18 236 Dirk-Philip Brandes singular value σ min ( D D), which is he square roo of he minimal eigenvalue of ( D D)( D D), is sricly posiive, and if [Cov(M (i),m(j) )]2 < 4 σ min( D D) 2, (49) hen (44) applies i,j= Proof: Assume ha E[U ] has only eigenvalues wih sricly negaive real pars Then so does D = E[U ] E[U ] by Fac 7 of Bernsein (2009) and hence also D D In paricular, D D is inverible so ha is minimal singular value is sricly posiive By Fac 87 of Bernsein (2009), he sum D = D +B has only eigenvalues wih sricly negaive real pars if B F < /2σ min ( D D), where F denoes he Frobenius norm Due o he form of B, we see immediaely ha B 2 F = p 2 j= b2 p 2,j, hence (49) Le us denoe wih A he column sum and wih A he row sum norm of a marix A, respecively Furher, κ (A) = A A and κ (A) = A A denoe he condiion number of an inverible A wih respec o and, respecively Proposiion 47 Suppose ha E C 2 < and ha E[U ] has only pairwise disinc eigenvalues wih sricly negaive real pars, which we denoe by µ,,µ p Le D and B be as in Proposiion 46, denoe µ µ p S :=, Λ := diag(µ,,µ p ), µ p µ p p and by spec( D) he se of all eigenvalues of D Then E[U ] and D are diagonalizable, more precisely S E[U ]S = Λ and (S S ) D(S S) = Λ Λ Furher, if (a) κ (S) 2 max (b) κ (S) 2 i,j=,,p Cov(M(i),M(j) Cov(M (i) i,j= hen (44) applies,m(j) ) < min Re(λ), or λ spec( D) ) < min Re(λ), λ spec( D) Proof: ThaE[U ]isdiagonalizableandhas E[U ]S = Λunderheassumpion of pairwise disinc eigenvalues, is well-known, see, for example, Fac 564 in Bernsein (2009) Then (S S ) D(S S) = (S S )(E[U ] I +I E[U ])(S S) = S E[U ]S S S +S S S E[U ]S = Λ I +I Λ = Λ Λ, so ha D is diagonalizable and has only eigenvalues wih sricly negaive real pars Le r {, } By he Theorem of Bauer-Fike (see Theorem 722 in

19 Coninuous ime ARMA processes wih random Lévy coefficiens 237 Golub and Van Loan, 996) we have for µ being an eigenvalue of D = D+B ha min λ spec( D) λ µ κ r (S S) B r In paricular, if κ r (S S) B r < min Reλ, λ spec( D) hen D can only have eigenvalues wih sricly negaive real pars Observe ha by Fac 996 in Bernsein (2009) i holds κ r (S S) = S S S r S r = S 2 r S 2 r = κ r(s) 2 so ha he saemen follows Remark 48 (a) Proposiion 47 can also be formulaed for oher naural marix norms corresponding o he r-norms wih r [, ], ie Ax A r = sup r x 0 x r (b) For r = observe ha Theorem in Gauschi (962) gives esimaes for he condiion number κ (S) when S has he form as in Proposiion 47 which hen gives feasible condiions for (44) o hold (c) Boh Proposiion 46 and 47 sae ha, if a sricly saionary CARMA process wih finie second momens and marix A whose eigenvalues have only sricly negaive real pars is given, hen an RC-CARMA process wih E[U ] = A can be chosen o be sricly saionary wih finie second momens, provided he variances of he M (i) are sufficienly small In oher words, he CARMA marix may be slighly perurbed and sill give a sricly saionary RC-CARMA process wih finie second momens Example 49 Consider an RC-CARMA(2, ) process under he assumpion of Proposiion 47 Denoe wih λ λ 2 he eigenvalues of E[U ] wih sricly negaive real pars If where κ max i,j=,2 Cov(M(i),M(j) ) < min λ spec( D) Re(λ), ( ) 2 (+max{ λ, λ 2 }) max{2, λ + λ 2 } κ =, λ 2 λ hen D+B has only eigenvalues wih sricly negaive real pars Proof: Since E[U ] is a companion marix, we have [ ] [ S = and S λ λ = 2 Hence, sraighforward calculaions yield λ 2 λ 2 λ λ 2 λ λ λ 2 λ λ 2 λ κ (S) = (+max{ λ, λ 2 }) max{2, λ + λ 2 } λ 2 λ Observe ha κ (S) = κ (S) such ha Proposiion 47 (b) gives a weaker sufficien condiion ]

20 238 Dirk-Philip Brandes Le X = (X ) R be a weakly saionary real-valued sochasic process wih E X 2 < for each R, hen he auocovariance funcion of X wih lag h is defined by γ X (h) := Cov(X +h,x ) = E[(X +h E[X +h ])(X E[X ])], h R and he auocorrelaion funcion of X is ρ X (h) := γ X(h) γ X (0) = Corr(X +h,x ), h R If γ X : R R is he auocovariance funcion of such a process X = (X ) R wih R γ X(h) dh <, hen is Fourier ransform f X (ω) := e iωh γ X (h)dh, ω R, 2π is called he specral densiy if he inegral exiss I is well-known (eg Brockwell, 200a) ha he specral densiy of a CARMA (p,q)-process S = (S ) 0 of order q < p is given by f S (ω) = σ2 b(iω) 2 2π a(iω) 2, ω R, wih σ 2 being he varianceof he driving Lévy process Under he saed condiions of Theorem 45 we see ha he specral densiy of an RC-CARMA(p,q) process R = (R ) 0 wih parameers C and b is given by f R (ω) = f S (ω) RC, ω R, wheref S (ω) denoes he auocovariancefuncion ofhe associaedcarma process wih A = E[U ] as in Theorem 45 and he consan RC is as in (47) Remark 40 I is well-known ha if S = (S ) R is a weakly saionary CARMA (p,q) process, hen he equidisanly sampled process S = (S n ) n N0, > 0, is a weak ARMA(p,q ) process for some q < p, see eg Secion 3 in Brockwell (2009) Since under he condiions of Theorem 45 he auocovariance funcion of an RC- CARMA(p,q) processr = (R ) 0 differsfromhaofhe associaedcarma(p,q) process only by a muliplicaive consan, i follows immediaely ha also R = (R n ) n N0 is a weakly saionary ARMA(p,q ) process for some q < p Nex, we evaluae exemplarily he covariance srucure of an RC-CARMA(2, ) process under he assumpion E[L ] = 0 Example 4 (Covariance of RC-CARMA(2,)) Le R = (R ) 0 be an RC-CARMA(2,) process wih parameers C = (C ) 0 = (M (),M (2),L ) 0, b and sricly saionary sae vecor process V = (V ) 0 Le V 0 be independen of C and C a semimaringale wih respec o he given filraion F Assume ha E[L ] = 0, ha (4) holds for κ = 2, and denoe wih S = (S ) 0 he associaed CARMA process characerized by Theorem 45 Then, for all 0, Cov(R +h,r ) = Cov(S +h,s ) RC = b e he[u] [ b0 E[M (2) ] b for all h 0, where RC = 2E[M () ]E[M(2) ] (2E[M () ] Var[M() ])E[M(2) ] Var[M(2) ] ] E[L 2 ] 2E[M () ] RC (40)

21 Coninuous ime ARMA processes wih random Lévy coefficiens 239 Proof: Under he assumpions made, an applicaion of Theorem 45 yields an associaed CARMA process S and he firs equaliy in (40) Clearly, [ ] 0 E[U ] = E[M (2) ] E[M() ], and he general form of Cov(R +h,r ) is obained from Proposiion 43, ie Cov(R +h,r ) = b e he[u] vec ( D e 4 )E(L 2 )b (4) Easy calculaions show ha 0 0 E[M (2) D = ] E[M() ] 0 E[M (2) ] 0 E[M() ], Var[M (2) ] η η Var[M() ] 2E[M() ] where η := Cov[M (2),M() ] E[M(2) ] Le = (2E[M () ] Var[M() ])E[M(2) ] Var[M(2) ], [ ] and denoe y = 0 0 E[M (2) ] Then Dy = e4 such ha [ ] vec ( D e 4 ) = vec = 0 E[M (2) E[M (2) ] ] Summarizing, vec ( D e 4 )b = = [ E[M (2) ] 0 0 ] [ b0 E[M (2) ] b and wih (4) we ge he saed shape of (40) ] [b0 ] () 2E[M ]E[M(2) ] b 2E[M () ] 2E[M () ] RC, Observe ha for M () and M (2) being deerminisic, RC =, hence dividing (40) by RC gives he auocovariance funcion of a CARMA(2,) process Example 42 (Covariance of RC-CARMA(3,2)) Le R = (R ) 0 be an RC-CARMA(3,2) process wih parameers C = (C ) 0, b and sricly saionary sae vecor process V = (V ) 0 Le V 0 be independen of C and C a semimaringale wih respec o he given filraion F Assume ha E[L ] = 0 and (4) hold for κ = 2 Then, for all,h 0, Cov(R +h,r ) = b e he[u] E[M () ] E[M (2) ]b 0 b 2 b E[M (2) ]b 2 b 0 E[L 2 ] 2(E[M () ]E[M(2) ] E[M(3) ]) RC,

22 240 Dirk-Philip Brandes where RC = 2(E[M () ]E[M(2) ] E[M(3) ]) ξ +2(Cov(M (3),M() ) E[M(3) wih ξ = (2E[M () ] Var[M() ])E[M(2) ] Var[M(2) ] E[M() ]) ] E[M (3) ] ]Var[M(3) The proof follows by similar calculaions as in Example 4 and is lef o he reader 5 Simulaions We compare in his secion wo simulaions of an RC-CARMA(2, ) process, one when E[U ] has only real sricly negaive eigenvalues and he oher when E[U ] has complex eigenvalues wih sricly negaive real pars For our simulaions we have chosen as random coefficien processes M (), M (2) wo independen compound poisson processes, ie wih depicion N () M () = i= N (2) X () i and M (2) = i= X (2) i, 0 In he case of real eigenvalues we have chosen E[N () ] = 5, E[N(2) ] = 2, X() i N(,03 2 ), and X (2) i N(025,02 2 ), and as driving process a sandard Brownian moion B = (B ) 0 Hence, for D = E[U ] E[U ] and D = D +E[U U ] E[U ] E[U ] we have E[U ] = [ ] 0, D = /2 3/2 0 0 /2 3/2 0 /2 0 3/2, and 0 /2 / D = /2 3/2 0 /2 0 3/ /2 /2 365 Consequenly, we have for E[U ] he eigenvalues µ = /2 and µ 2 =, and for D he eigenvalues λ 029, λ i, λ 3 = 29 28i, and λ 4 = 5 Since all eigenvalues of D have sricly negaive real pars, we obain he exisence of a sricly saionary soluion by Remark 42 and Remark 44 Neverheless, observe ha Var(M () )+Var(M(2) ) = 84 4 σ min( D D) showing ha he condiion in Proposiion 46 is no necessary Moreover, so ha ( ) 2 (+max{ λ, λ 2 }) max{2, λ + λ 2 } κ = = 25 λ 2 λ κ max i=,2 Var(M(i) ) = min λ spec( D) Re(λ) =,

23 Coninuous ime ARMA processes wih random Lévy coefficiens 24 showing ha also he condiion in Example 4 and hence in Proposiion 47 is no necessary In case of complex eigenvalues we have chosen E[N () ] = 2, E[N(2) ] = 75, X () i N(05,0 2 ), and X (2) i N(04,005 2 ), and have lef he driving process unchanged Then [ ] 0 E[U ] = 3 gives wo complex-valued eigenvalues µ i and µ 2 = 05 66i Also, D has jus eigenvalues wih sricly negaive real pars Furhermore, an observaion similar o he one above can be made showing non-necessiy of he condiions in Proposiion 46 and model ACVF of RC-CARMA sample ACVF based on R(00),,R(00,000) sample ACVF based on R(00),,R(00) model ACVF of CARMA RC-CARMA(2,) ACVF Time (a) Simulaion wih eigenvalues of E[U ] chosen o be µ = 2 and µ 2 = Lag (b) ACVFs wih eigenvalues of E[U ] chosen o be µ = 2 and µ 2 = model ACVF of RC-CARMA sample ACVF based on R(00),,R(00,000) sample ACVF based on R(00),,R(00) model ACVF of CARMA 5 04 RC-CARMA(2,) ACVF Time (c) Simulaion wih eigenvalues of E[U ] chosen o be µ 05+66i and µ 2 = 05 66i Lag (d) ACVFs wih eigenvalues of E[U ] chosen o be µ 05+66i and µ 2 = 05 66i Figure 5 Simulaed RC-CARMA(2,) process and ACVFs Forbohhecomplexandherealeigenvaluescase,wehavesimulaed0,000,000 observaions wih a mesh size k = 00, ie R 00,R 002,,R 00,000 In Figure

24 242 Dirk-Philip Brandes 5(a) and (c) we see he corresponding plos unil ime 300 Plos 5(b) and (d) show he corresponding auocovariance funcions (ACVF) The solid line corresponds o he model auocovariance funcion, he dashed one o he sampled ACVF based on he daa R 00,,R 00,000, and he doed shows he model ACVF of he corresponding CARMA(2, ) process The dashed-doed line shows he sample ACVF based on R 00,,R 00 We see ha using 0, 000, 000 observaions o calculae he sample auocovariance funcion, i nearly agrees in boh cases wih he model auocovariance funcion Plo (d) shows a sinusoidal oscillaion which is also in he CARMA case characerisic for allowing complex eigenvalues and visualizes he variey of possible auocovariance funcions RC-CARMA(2,) 5 0 CARMA(2,) Time Time 0 4 (a) Simulaion of RC-CARMA(2, ) wih eigenvalues of E[U ] chosen o be µ = 2 and µ 2 = (b) Simulaion of associaed CARMA(2, ) wih A = E[U ] Figure 52 Simulaed RC-CARMA(2,) and CARMA(2,) processes Figure 52(a) shows all simulaed observaions of he RC-CARMA(2, ) process where E[U ] has real eigenvalues On he oher hand, Figure 52(b) shows an equally sized simulaion of he associaed CARMA(2,) process, ie wih he choice A = E[U ] I can be seen ha he RC-CARMA(2,) process provideslarger ouliers around he beween 5 and 5 concenraed band han he CARMA(2, ) process around is band This may indicae possible heavy ails of RC-CARMA processes We jusify hese observaions inuiively in he following remark recalling resuls on random recurrence equaions Remark 5 Observe ha a saionary CARMA process driven by a Brownian moion has a normal marginal saionary disribuion, in paricular, i has ligh ails On he oher hand, as a consequence of resuls of Kesen (973) and Goldie (99), i is known ha a generalized Ornsein-Uhlenbeck process and so an RC- CARMA(,0) process will have Pareo ails under wide condiions, even if he driving process is a Brownian moion, see eg Lindner and Maller (2005) (Theorem 45) or Behme (20) (Theorem 4)

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