Asymptotic Properties of an Estimator of the Drift Coefficients of Multidimensional Ornstein-Uhlenbeck Processes that are not Necessarily Stable

Asymptotic Prop erties of an Estimato r of the Drift Co efficien ts of Multi d imensional Ornstein-Uhlen b ec k Pro cesses that are not Necessarily Stable Gopal K. Basak ∗ and Philip Lee † Abstract In this pap er, we in vestigate the consistency and asymptotic effi- ciency of an estimator of the dr if t matrix, F , o f Ornstein-Uhlen b ec k pro cesses that are not necess a rily stable. W e conside r all the cases. (1) The eigenv alues of F are in the r ig h t half spa c e (i.e., eigenv alues with po sitiv e real par ts). In this c ase the pro cess g ro ws e x ponentially fast. (2) The e igen v alues of F are on the left half space (i.e., the eigenv alues with negative or zero r eal parts). The pro cess wher e all eigenv alues of F hav e negative real parts is called a stable pro cess and has a unique inv ariant (i.e., stationar y) distr ibut ion. In this case the pro cess do es not grow. When the eig en v alues of F have zero r eal parts (i.e., the case of zer o eigenv alues and pur e ly imaginar y eigen v alues) the pro - cess g ro ws p olynomially fast. Co nsidering (1) and (2) separately , we first show that an estimator, ˆ F , of F is c onsisten t. W e then co mbine them to present results for the general Or nstein-Uhlen b ec k pro cesses. W e adopt similar pro cedure to show the asymptotic efficiency of the estimator. Key words and phrases: Ornstein-Uhlenb eck pr ocesses, stable pro cess, drift co efficien t m a trix, estimation, consistency , asymptotic efficiency . AMS sub ject classification: 62M05 (60F15) ∗ (Corresponding auth o r) Stat-Math Unit, Indian Statistical In s titute, Kolk ata 700 108, India. E-mail: gkb@isical.ac.in † JPMorganChas e Bank, N.A., Asia Rates Strategy , Hong Kong. Email: philip.pk.lee@jpmorgan.com 1 1 In tro duction Multidimensional p rocesses with linear drift parameter ha ve b een used for mo delling v arious ph ysical phenomena. Among recent pap ers, works b y Jankunas and Khasmin skii ([12]) a nd Khasminskii, Krylo v a nd Moshc h u k ([15]) on the estimatio n of the dr i ft parameters of linear stoc hastic differ- en tial equations (of the f o r m, dX t = AX t dt + P n i =1 σ i X t dw i ( t ) and dX t = A θ X t dt + P m i =1 σ i X t dw i ( t )) can b e men tioned. It should b e noted that our w ork on Orn st ein-Uhlen b ec k (OU) pr ocesses do es not follo w from theirs and that the m etho dology used in our pap er is also quite different from theirs. The motiv ation for this w ork comes fr o m Lai and W ei’s pap er [20], in whic h the authors hav e shown the strong consistency of the least square estimators of the co efficien ts of the discrete univ ariate general AR(p) pro cesses. In this pap er, w e not only sho w that an estimator (whic h is the maxim u m like liho od estimator in the sp ecial case when A is nonsin g ular) of th e drift p a rameter of the general m ultidimensional OU pr o cess is consisten t b ut also sho w that it is asymptotically efficient . W e consider the follo wing SDE r e presen tation of the OU pro ce ss: d Y t = F Y t dt + AdW t (1.1) with an y s tarting p oin t Y 0 indep enden t of the Bro wn ia n motion { W t , t ≥ 0 } . Here Y is a p -dim en sio n al pro cess, A is a constan t matrix of p × r dimesnion and W t is a r -dimensional standard Brownian motio n . Notice that it is alw a ys ea sier to e stimate A through qu a dratic v ariation of the pr o cess b y using Itˆ o’s ru le . But, estimating F is usu a lly the more difficult task. It is generally b eliev ed that one needs stationarit y of the pro cess to estimate F . Ho wev er, one ma y observe, R T 0 d Y t Y ′ t = F ( R T 0 Y t Y ′ t dt ) + A ( R T 0 dW t Y ′ t ). Th us, w e define, ˆ F T = ( R T 0 d Y t Y ′ t )( R T 0 Y t Y ′ t dt ) − 1 = F + A ( R T 0 dW t Y ′ t )( R T 0 Y t Y ′ t dt ) − 1 when ( R T 0 Y t Y ′ t dt ) is inv ertible and, in this case, the estimator is un biased 2 (as the exp ec tation of the second term is zero). W e sho w here that ˆ F T is a consisten t and an asymptotically efficien t estimator of F , irresp ectiv e of the stationarit y (or stabilit y) of the pro cess, pro v id e d F and A toget her satisfy a RANK condition (a), giv en in Section 2. This RANK condition is essen tial to pro ve that ( R T 0 Y t Y ′ t dt ) is in vertible. W e note here, if A is a non s i ngular matrix, the RA NK condition automatic ally holds. In fact, it is also easy to see that for a con tin u o us autoregressiv e pro cess (i.e., CAR(p)), th e RANK condition holds. W e a lso mak e another assumption, c ondition (b). It is the distinctness of the eige n v alues with p ositiv e real parts. Ho wev er, w e p oi n t out that this condition can b e relaxed with a condition (b’) and also th at if none of the conditions (b) or (b’) hold it is s till p ossible to p roceed with th e estimatio n (see the discussion after R emark 3.2). Notice that the condition (b’) holds for the d r ift F in CAR(p) p rocesses. The estimation of parameters for the stochastic pro cesses ha v e extensivel y studied (see for example, F eigin [8], Basa wa, F eigin and Heyde [6], Basa wa and Prak asa Rao [5], Dietz and Ku t o ya n ts [7], Kuto y ants [17, 18], Barnd o rff- Nielson and Sorensen [2 ], K uto yan ts and Pilib ossian [19], Janku nas and Khasminskii [12], Khasminskii, Kr ylo v and Moshch uk [15] Prak asa Rao [23, 24] and references therein). Therefore, the estimation of th e paramater and its asymp to tic stu dies ha ve not b ee n new. Ho w ev er, as far as we kno w, full study of m ultidimens i onal OU pro cesses parameter estimation and the study o f its asymp to tics ha v e not b een done for the mixed mo del. Ap art from sh o win g consistency and asymp totic efficiency for the multidimensional (matrix v alued) v ariable that do es not follo w fr o m that of un i v ariate or v ector v alued case (see, for example, Kaufmann [14], W ei [25], Basa w a and Prak asa Rao [5], Diet z and Ku t o ya n ts [7], Kuto y an ts [17, 18], Barnd o rff- Nielson and Sorensen [2], Prak asa Rao [23, 24] and references therein) it also 3 dev elops new method o logy to deal with suc h cases as is don e in Kaufm a nn [14] and W ei [25]. Our pap er is o rganized as follo ws. In Sectio n 2, w e present the basic as- sumptions and the mai n theorems. In Sectio n 3, we d esc rib e the case in whic h the eigen v alues of F h a ve p ositiv e real parts. Method ol ogy u sed here is similar to that of Lai and W ei’s paper [20], wh il e the case in which the eigen v alues of F ha ve negativ e or zero real p a rts is quite different from them and it is d i scussed in Section 4. This case, in fact, combines the three cases, zero eigen v alues, purely im aginary eigen v alues and the eigen v alues with negativ e real parts. De tails on the rates of gro w t h and so forth for zero eigen v alues and imaginary eigen v alues are giv en in the App endix. Secti on 5 examines the m i xed case for consistency . The section 6 p rese n ts the results on asymptotic efficiency and some concluding r e marks. 2 Basic A s su mpti ons and the M ain Theorem W e can decomp ose any p × p matrix F into the rational canonical form M F = GM =   G 0 0 0 G 1     M 0 M 1   where G i are p i × p i matrices and M i are p i × p matrices f or i = 0 , 1 and p 0 + p 1 = p . Rows of M i and ro ws of M j are orthogonal f or i 6 = j . All ro ots of G 0 lie in the right h a lf space; all ro ots of G 1 lie on the left half space. 4 EXAMPLE Let A =            2 − 1 0 1 0 0 − 8 6 14 1 0 10 − 4 − 1 4 − 1 0 − 10 6 16 1 0 − 5 3 7 0            . Then the charac teristic p olynomial of A is f ( t ) = ( t − 2) 3 ( t 2 + 1) . Th us φ 1 ( t ) = t − 2 and φ 2 ( t ) = t 2 + 1 are the distin ct irreducible monic divisors of f ( t ). After computation, we find that g ( t ) = φ 1 ( t ) 2 φ 2 ( t ) = ( t − 2) 2 ( t 2 + 1) is the minimal p olynomial of A and thus th e companion matrices for φ 2 1 ( t ) = ( t − 2) 2 and φ 1 ( t ) = t − 2 are given b y   0 − 4 1 4   and 2 . Similarly , the companion matrix for φ 2 ( t ) = t 2 + 1 is   0 − 1 1 0 .   The rational canonical form of A is thus H A =            0 − 4 0 0 0 1 4 0 0 0 0 0 2 0 0 0 0 0 0 − 1 0 0 0 1 0            In the example ab o v e, the rational canonica l form of A is formed by 3 blocks:   0 − 4 1 4   , 2 and   0 − 1 1 0   . Therefore the dimensions of the 3 blo c ks are 2, 1 and 2 resp ecti v ely . 5 ASSUMPTION ( a ) RANK h A : F A : · · · : F p − 1 A i = p. (2.1) (b) Th e eigenv alues of F , which h a ve p ositiv e real p a rts, are all distinct. Observe that, fr om (1.1) Y t = e F t Y 0 + R t 0 e F ( t − s ) AdW s and thus ha ve a m u lt iv ariate Gaussian distribution with the mean e F t and the co v ariance matrix R t 0 e F t AA ′ e F ′ t . Since Y t is Gaussian it has a p ositiv e density if and only if the co v ariance matrix is non s i ngular. Th e RANK assu mption whic h is the sp ecial case of H¨ ormander’s hyp oellipticit y condition ensures the p ositiv e densit y of Y t (for details, see [11]), and hence the nonsin g ularit y of cov ariance matrix. F ollo wing Basa wa and Rao ([5], pp .) it is clear th a t R T 0 Y t Y ′ t dt is n on s i ngular under the R ANK assumption. Let F A = [ A : F A : · · · : F p − 1 A ]. Then RAN K ( F A ) = p b y the RANK assumption. Consider for i = 0 , 1, p i = RANK( M i F A F − 1 A ) ≤ RANK( M i F A ) ≤ p i where F − 1 A is the righ t in v erse of F A . Therefore, RANK ( M i F A ) = p i for i = 0 , 1. Since M i F A = h M i h A : F A : · · · : F p − 1 A ii = h M i A : M i F A : · · · : M i F p − 1 A i = h M i A : G i M i A : · · · : G p − 1 i M i A i , and as the higher p o wer of G i can b e exp r e ssed as a linear com b inat ion of I , G i , . . . , G p i − 1 i , RANK h M i A : G i M i A : · · · : G p i − 1 i M i A i = RANK h M i A : G i M i A : · · · : G p − 1 i M i A i = p i . (2.2) 6 If we transf o rm the pro cess Y t to U it = M i Y t for i = 0 , 1, M i d Y t = M i F Y t dt + M i AdW t , i.e., dU it = G i U it dt + ( M i A ) dW t . F rom (2.2) and the argumen t giv en ab o v e, w e conclude that R T 0 U it U ′ it dt is p ositiv e d e finite a.s. for i = 0 , 1. W e no w presen t our m a in theorems wh o se p roofs are giv en in Section 5 and in Section 6, r e sp ectiv ely . Th roughout the pap er, w e use λ min ( C ) and λ max ( C ) to denote the minimum and maxim um eigen v alues of a matrix C . THEOREM 2.1 Supp ose, for the Ornstein-Uhlenb e ck pr o c ess define d in (1.1), the assumptions (a) and (b) hold. Define ˆ F T = ( R T 0 d Y t Y ′ t )( R T 0 Y t Y ′ t dt ) − 1 . Then lim inf T →∞ 1 T λ min Z T 0 Y t Y ′ t dt ! > 0 a . s . (2.3) and lim T →∞ ˆ F T = F a . s . THEOREM 2.2 Under the assumptions of The or em 2.1, it fol lows that E (T r [( ˆ F T − F ) E ( C T )( ˆ F T − F ) ′ ]) 1 / 2 = O (1) as T → ∞ , wher e ˆ F T is as define d in The or em 2.1 and C T =  R T 0 Y t Y ′ t dt  . 3 Eigen v alues in the Righ t Half Space W e consider the case where all the eigen v alues of F hav e p ositiv e real parts. In this case, it can b e seen that k Y t k → ∞ exp on entially fast as t → ∞ . 7 T o introdu ce the main result of this section w e define a Gauss ian r a ndom v ariable Z = Y 0 + Z ∞ 0 e − F s AdW s . Since all the eigen v alues of F hav e p ositiv e real parts, it is clear that, e − F t Y t = Y 0 + R t 0 e − F s AdW s con verge s a.s. to Z as t → ∞ . W e no w deriv e the follo wing results. THEOREM 3.1 In addition to the assumptions and notations of The or em 2.1, assume f urt her that r e al p arts of al l th e eigenvalues of F ar e p ositive. Then, e − F T Z T 0 Y t Y ′ t dt ! e − F ′ T con verge s a . s . to B = Z ∞ 0 e − F t ( Z Z ′ ) e − F ′ t dt. Mor e over, B is p ositive definite with pr ob ability 1. Conse quently, lim T →∞ T − 1 log λ min Z T 0 Y t Y ′ t dt ! = 2 λ 0 a . s . lim T →∞ T − 1 log λ max Z T 0 Y t Y ′ t dt ! = 2Λ 0 a . s . (3.1) Here and throughout the pap er, log x means the n a tural logarithm of x . Also, in the sequel w e shall let || x || denote the Euclidean norm of a p - dimensional v ector x = ( x 1 , · · · , x p ) ′ , i.e., || x || 2 = x ′ x . Moreo v er, by v iewin g a p × p matrix A 0 as linear op erator, we defin e || A 0 || = sup || x || =1 || A 0 x || . Th us, || A 0 || 2 is equal to the maxim um eigen v alue of A ′ 0 A 0 . Moreo v er, if A 0 is symmetric and non-negativ e d efi nite , then || A 0 || = λ max ( A 0 ). In particular, for the companion matrix e − F T in Th e orem 3.1, we ha ve the follo wing Lemma. LEMMA 3.1 Under the hyp othesis of The or em 3.1 log || e F T || ∼ log || e F ′ T || ∼ Λ 0 T , and log || e − F T || ∼ log || e − F ′ T || ∼ − λ 0 T (3.2) 8 wher e we use the notation f ( T ) ∼ C T k to denote lim T →∞ T − k f ( T ) = C . Pro o f. Supp ose Re[ λ k ( F )] > 0 for k = 1 , 2 , · · · , p . Th e n | e λ k ( F ) | = e Re[ λ k ( F )] > 1 for k = 1 , 2 , · · · , p. Let λ 0 = min 1 ≤ k ≤ p Re[ λ k ( F )], Λ 0 = max 1 ≤ k ≤ p Re[ λ k ( F )]. Denote the sp ec- tral radius of F by r σ ( F ) (cf. [16]). Then lim T →∞ || e F T || 1 T = r σ ( F ) = sup λ ∈ σ ( e F ) | λ | = exp h sup λ ∈ σ ( F ) Re( λ ) i = e Λ 0 and so log || e F T || ∼ log || e F ′ T || ∼ Λ 0 T . Similarly , log || e − F T || ∼ log || e − F ′ T || ∼ − λ 0 T since lim T →∞ || e − F T || 1 T = su p λ ∈ σ ( e − F ) | λ | = exp h sup λ ∈ σ ( − F ) Re( λ ) i = e − λ 0 . Th us, we h a ve th e pr oof of Lemma 3.1 Pro o f of T heo rem 3.1. Let Z t = Y 0 + R t 0 e − F s AdW s , then Y t = e F t Z t and Z t con verge s a . s . to Z = Y 0 + Z ∞ 0 e − F s AdW s . Let B T = R T 0 e − F t Z T Z ′ T e − F ′ t dt ,           e − F T Z T 0 Y t Y ′ t dt ! e − F ′ T − B T           =           Z T 0 e − F ( T − t ) Z t Z ′ t e − F ( T − t ) dt − Z T 0 e − F t Z T Z ′ T e − F ′ t dt           =           Z T 0 e − F t  Z T − t Z ′ T − t − Z T Z ′ T  e − F ′ t dt           ≤ Z T 0 || e − F t || || e − F ′ t || ( || Z T − t || + || Z T || ) || Z T − Z T − t || dt = Z T / 2 0 || e − F t || 2 ( || Z T − t || + || Z T || ) || Z T − Z T − t || dt + Z T T / 2 || e − F t || 2 ( || Z T − t || + || Z T || ) || Z T − Z T − t || dt. (3.3) 9 Since Z t con verge s almost surely to a finite rand o m v ariable Z , sup { t ≥ 0 } k Z t k is fin ite almost su r e ly and for eac h t ≥ T / 2, || Z T − Z T − t || , b eing a cauc hy sequence, con v erges to zero, almost sur e ly , as T → ∞ . Also, b y Lemma 3.1, R ∞ 0 || e − F t || 2 dt < ∞ . Th us, we ge t, ∀ ω outside a n ull set, ∀ ǫ > 0, there exists a T 0 ( ω ) suc h that k Z t ( ω ) − Z ( ω ) k < ǫ/ (1 + R ∞ 0 || e − F t || 2 dt + 2 sup { t ≥ 0 } k Z t ( ω ) k ) for all t ≥ T 0 ( ω ). Fixing one such ω , for T ≥ 2 T 0 ( ω ) w e ha ve the first in tegral of (3.3), which is less than ǫ and the second in tegral go es to zero as sup { t ≥ 0 } k Z t ( ω ) k is finite an d R T T / 2 || e − F t || 2 dt → 0 as T → ∞ . Let B = R ∞ 0 e − F t Z Z ′ e − F ′ t dt , then with pr o b a bilit y 1, || B T − B | | ≤ Z ∞ T || e − F t Z Z ′ e − F ′ t || dt + Z T 0 || e − F t ( Z Z ′ − Z T Z ′ T ) e − F ′ t || dt ≤ || Z Z ′ || Z ∞ T || e − F ′ t || || e − F t || dt + || Z Z ′ − Z T Z ′ T || Z T 0 || e − F t || || e − F ′ t || dt → 0 a . s ., as T → ∞ . (3.4) Therefore, e − F T Z T 0 Y t Y ′ t dt ! e − F ′ T con verge s a . s . to B = Z ∞ 0 e − F t Z Z ′ e − F ′ t dt. (3.5) T o sh ow B = R ∞ 0 e − F t Z Z ′ e − F ′ t dt is p ositiv e d e finite with probabilit y 1, observ e that Z has p ositiv e Gaussian density . Hence P ( Z 6 = 0) = 1. Fix an ω , suc h that Z ( ω ) 6 = 0. Sup pose, if p ossible, x ′  Z ∞ 0 e − F t Z ( ω ) Z ( ω ) ′ e − F ′ t dt  x = 0 for some nonzero vect or x ∈ R p . Then, for almost all t ∈ (0 , T ), x ′ e − F t Z ( ω ) = 0, i.e., f o r almost all t ∈ (0 , T ), ∞ P k =0 1 k ! ( − 1) k x ′ F k t k Z ( ω ) = 0. Th is implies x ′ F k Z ( ω ) = 0 , for k = 0 , 1 , · · · , p − 1. By the a ssumption (b), P p − 1 k =0 a k F k is n o nsingular for any real num b er a k with not all of them b eing zero. Hence, for any nonzero 10 v ector in R p , in particular for x , x ′ P p − 1 k =0 a k F k is a nonzero v ector. In other wo rds, for nonzero vec tor x , P p − 1 k =0 a k ( x ′ F k ) is nonzero for an y nonzero v ector ( a 0 , . . . , a p − 1 ). Th us          x ′ x ′ F . . . x ′ F p − 1          is a nonsingular matrix. Hence,          x ′ x ′ F . . . x ′ F p − 1          Z ( ω ) = 0 implies Z ( ω ) = 0, which is a con tr ad iction. Th u s, w e arr ive at a co n tradiction since Z has a p ositiv e Gaussian d e nsit y an d hence Z ca nnot b e equal to zero on a set of p ositiv e measures. Therefore, w e conclude that B is p ositiv e d e finite with p robabilit y one. T o pro ve (3.1), we state the follo wing elemen tary results (for the pro of, s e e Lemma 2 of [20]): LEMMA 3.2 L et A , C b e p × p matric es suc h that C is symmetric and non- ne g ative definite. Then λ max ( C ) λ max ( AA ′ ) ≥ λ max ( AC A ′ ) ≥ λ min ( C ) λ max ( AA ′ ) , λ max ( C ) λ min ( AA ′ ) ≥ λ min ( AC A ′ ) ≥ λ min ( C ) λ min ( AA ′ ) . . W e con tin ue the p roof of (3.1) of Theorem 3.1. F r o m Lemma 3.2 we get, log λ min Z T 0 Y t Y ′ t dt ! ≤ log λ max " e − F T Z T 0 Y t Y ′ t dt ! e − F ′ T # − log λ max  e − F T e − F ′ T  ∼ 2 λ 0 T . 11 Also, log λ min Z T 0 Y t Y ′ t dt ! ≥ log λ min " e − F T Z T 0 Y t Y ′ t dt ! e − F ′ T # + log λ min  e F T e F ′ T  ∼ 2 λ 0 T . Therefore lim T →∞ 1 T log λ min Z T 0 Y t Y ′ t dt ! = 2 λ 0 a . s . On the other hand, log λ max Z T 0 Y t Y ′ t dt ! ≤ log λ max " e − F T Z T 0 Y t Y ′ t dt ! e − F ′ T # + log λ max  e F T e F ′ T  ∼ 2Λ 0 T . Also, log λ max Z T 0 Y t Y ′ t dt ! ≥ log λ min " e − F T Z T 0 Y t Y ′ t dt ! e − F ′ T # − log λ min  e − F T e − F ′ T  ∼ 2Λ 0 T . Therefore lim T →∞ 1 T log λ max Z T 0 Y t Y ′ t dt ! = 2Λ 0 a . s . Hence, we ha ve the pro of of Th eo rem 3.1. COROLLARY 3.1 Under the same assumptions and notations as in The o- r em 3.1, ( i ) lim T →∞ Z T 0 || e − F T Y t || dt = Z ∞ 0 || e − F t Z || d t < ∞ a . s . (3.6) ( ii ) 1 √ T Z T 0 dW t Y ′ t ! e − F ′ T = O ( T − 1 / 2 ) . 12 Pro o f. (i) Give n ǫ > 0 , ∀ ω outside a n u l l set, ∃ T 0 ( ω ) suc h that || Z t − Z || < ǫ ∀ t ≥ T 0 ( ω ) . F or T > T 0 ( ω ),      Z T 0 || e − F ( T − t ) Z t || dt − Z T 0 || e − F ( T − t ) Z || d t      ≤ Z T 0 || e − F ( T − t ) Z t − e − F ( T − t ) Z || d t ≤ Z T 0 || e − F ( T − t ) || || Z t − Z || dt ≤ Z T 0 ( ω ) 0 || e − F ( T − t ) || || Z t − Z || dt + Z T T 0 ( ω ) || e − F ( T − t ) || || Z t − Z || dt. As T → ∞ , the first term tend s to 0 since || e − F ( T − t ) || → 0. The second term also tends to 0 s i nce Z t → Z and R T T 0 ( ω ) || e − F ( T − t ) || dt ≤ R T 0 || e − F ( T − t ) || dt = R T 0 || e − F t || dt ≤ R ∞ 0 || e − F ( T − t ) || dt , whic h is fi nite. Therefore, lim T →∞ Z T 0 || e − F T Y t || dt = lim T →∞ Z T 0 || e − F ( T − t ) Z t || dt = lim T →∞ Z T 0 || e − F ( T − t ) Z || d t = Z ∞ 0 || e − F t Z || d t, whic h is finite almost surely , by Lemma 3.1. (ii) Let M t =  R t 0 dW s Y ′ s  e − F ′ T , which is a square inte grable martingale for 0 ≤ t ≤ T , with quadratic v ariation, < M > t = e − F T  Z t 0 Y s Y ′ s ds  e − F ′ T = e − F T C t e − F ′ T where C t = R t 0 Y s Y ′ s ds . By Karatzas and Shreve (cf [13] p174),  Z t 0 dW s Y ′ s  e − F ′ T = M t = B t = O  λ max  e − F T C t e − F ′ T  q ln ln λ max ( e − F T C t e − F ′ T )  = O (1 ) 13 since for t ≤ T , || e − F T C t e − F ′ T || ≤ || e − F T C T e − F ′ T || → B , almost surely , as T → ∞ and B = O (1). Therefore, 1 √ T Z T 0 dW t Y ′ t ! e − F ′ T = O ( T − 1 / 2 ) This completes the p ro of of Corollary 3.1 . REMARK 3.1 If all the eigen v alues of F h a ve p ositiv e real p arts, w e can relax condition (b) by ( b ′ ) p − 1 X k =0 a k F k b eing nonsingular for an y reals a 1 , . . . , a n with at least one of them b eing nonzero . (3.7) Notice that (b’) could hold ev en if all the eigen v alues o f F are equal (say , λ 0 ), but the degree of the minimal p olynomial of F and the d egree of the c haracteristic p olynomial of F are equal. REMARK 3.2 Supp ose, assumption (b) do es not hold. One c an stil l estimate the eigenvalues of F . Let the c haracteristic p olynomial of F be giv en as φ F ( x ) = a 0 Π k i =1 ( x − λ i ) p i Π l j =1 ( x 2 + b j x + c j ) q j where λ i are the real ro ots of multiplicit y p i and x 2 + b j x + c j are the irreducible p olynomials giving the complex ro ots with m u ltiplicit y q j and a 0 is a constant. Let the minimal p olynomial of F b e giv en b y ψ F ( x ) = Π k i =1 ( x − λ i ) r i Π l j =1 ( x 2 + b j x + c j ) s j with r i ≤ p i and s j ≤ q j . If r i = p i and s j = q j for all i, j , then the degree of the minimal p olynomial of F and the degree of the c haracteristic p olynomial of F are the 14 same and the assump tion (b’) h olds and our results follo w. If s ome of the r i s are less than p i s an d /or s j s are less than q j , then, (b’) do es not hold for F . How ev er, in that case, one can trans f orm F in the rational canonoical form as                   J 1 . . . J k K 1 . . . K l L                   F =                   B 1 · · · 0 0 · · · 0 0 0 . . . 0 0 · · · 0 0 0 · · · B k 0 · · · 0 0 0 · · · 0 C 1 · · · 0 0 0 · · · 0 0 . . . 0 0 0 · · · 0 0 · · · C l 0 0 · · · 0 0 · · · 0 D                                     J 1 . . . J k K 1 . . . K l L                   =                   B 1 J 1 . . . B k J k C 1 K 1 . . . C l K l D L                   where J i , K j and L are rectangular matrices of full row r an k , ( p i − r i ), ( q j − s j ), ( P i r i + P j s j ), resp ectiv ely , and D is a sq u are matrix of the dimens ion the s ame as the degree of the minimal p olynomial of F (i . e ., same as ( P i r i + P j s j )). F or eac h j , C j is a partitioned diagonal matrix (i.e., only the diagonal b lo c ks are nonzero blo c ks ), eac h blo c k is of dimen s ion 2 × 2, and its diagonal blo ck matrices are ident ical and rep eating exactly ( q j − s j ) times and ha ve the c haracteristic p olynomial x 2 + b j x + c j , and, for eac h i , B i is a d iagonal matrix w ith diagonal entries consisting of the real charact eristic ro ot λ i rep eating exactly ( p i − r i ) times. Th us, w e can w ork with D instead of F . F or D th e assumption (b’) holds, since th e degree of minimal p olynomial of D is same as that of F and, consequently , the degree of the min imal p olynomial of D is the same as the degree of the c haracteristic p olynomial of D . Estimation of D can b e done using th e SDE of LY t . F or B i and C j , one can consider eac h one separately and transform Y t to J i Y t and K j Y t and use th e SDE of an y comp onen t of J i Y t (as it h as the Mark o v prop er ty) to estimate λ i and the S DE of the first tw o (or, an y (2m-1)th and 2m th) comp onent s of K j Y t together, as they ha ve the Mark o v p rop erty , to estimate 15 a diagonal b lo c k of C j . Hence the assertion in the last remark. 4 Eigen v alues on the L eft Half Space In this Section, we study the asymptotic b eha v ior of OU pro cesses where the real parts of all the eigen v alues of F are either zero or negativ e. Unlik e the exp onentia l rate of g ro wth for || Y T || , λ max ( R T 0 Y t Y ′ t dt ), λ min ( R T 0 Y t Y ′ t dt ) in Theorem 3.1 and Corollary 3.1 for the the pro cess w here all the eigen v alues of F ha v e p ositiv e real parts, the follo w ing theorem sh o ws that these qu antitie s gro w at most p olynomially fast in t for these pro cesses. F or stable pro cesses Y t (i.e., eigen v alues of F w ith negativ e real parts), w e kno w from Basak and Bhattac h ary a [4] th at | Y x t − Y 0 t | → 0 a . s . as t → ∞ . Therefore, the prop erty of Y t starting at x is the same as that f rom 0. Hence, without loss of generalit y , we can assume that Y 0 = 0. THEOREM 4.1 Supp ose, for the Ornstein-Uhlenb e ck pr o c ess define d in (1.1), the RANK c ondition (2.1) holds and al l the eigenvalues of F have ne gative r e al p arts. Then lim inf T →∞ 1 T λ min Z T 0 Y t Y ′ t dt ! > 0 a . s . (4.1) Mor e over, λ max Z T 0 Y t Y ′ t dt ! = O ( T ) a . s . (4.2) 16 Pro of. T o pr ov e (4.1) and (4 .2), consider eac h component Y i t , Y j t of Y t , i, j = 1 , · · · , p . Let π b e the inv ariant distribution of Y . Then b y th e S tr ong La w of Large Numb ers, 1 T Z T 0 Y i t Y j t dt → E π ( Y i Y j ) < ∞ as T → ∞ , whic h follo ws, afortiori, by th e Law of the Iterated Logarithm b y Basak [3]. Therefore, 1 T Z T 0 Y t Y ′ t dt → E π ( Y Y ′ ) = Z ∞ 0 e F u AA ′ e F ′ u du, whic h is p ositiv e d efi nite a.s. Th erefore, lim inf T →∞ 1 T λ min Z T 0 Y t Y ′ t dt ! > 0 a . s . and λ max Z T 0 Y t Y ′ t dt ! = O ( T ) a . s . Hence, the pr o of. REMARK 4.1 (i) It is not difficult to see that f or stable Y t , f or any m ≥ 1, E h sup k − 1 ≤ t ≤ k ( Y ′ t P Y t ) m i is b ounded uniformly o v er k. Hence , it would follo w , for any δ > 0, || Y t || = o ( t 1 2 m + δ ) a.s. (ii) On th e other h and, since Y t → Y in distribution and Y is finite with probabilit y one, one obtains Y t = O p (1). COROLLARY 4.1 With the same notations and assumptions as in The or em 4.1, let C T = R T 0 Y t Y ′ t dt . Then ( i ) || C − 1 / 2 T || = O ( T − 1 / 2 ) , a . s . ( ii ) lim T →∞ Y ′ T C − 1 T Y T = 0 a . s . 17 Pro of. (i) Since lim inf T →∞ 1 T λ min ( C T ) > 0 a.s. from (4.1), th erefore || C − 1 / 2 T || 2 = λ max ( C − 1 T ) = 1 λ min ( C T ) = O ( T − 1 ) a . s . (ii) By the p revious remark 4.1 (i), w e note that, || Y ′ T C − 1 T Y T || ≤ || Y T || 2 || C − 1 T || = o ( T 1 / 2+2 δ ) O ( T − 1 ) a . s ., for some δ > 0 , small = O ( T − 1 / 2+2 δ ) Hence, the pr o of. THEOREM 4.2 Supp ose eig envalues of F have either ne gative or zer o r e al p arts (i.e., the e i genvalues ar e on the L eft Half Sp ac e, which includes zer o eigenvalues, pur ely i maginary eige nv alues, eig envalues with ne gative r e al p arts). Then, lim T →∞ Y ′ T Z T 0 Y t Y ′ t dt ! − 1 Y T = 0 a . s . T o p ro ve Theorem 4.2, we need the follo w ing lemma: LEMMA 4.1 L et ǫ > 0 ; define F ǫ = F − ǫI and d Y ǫ t = F ǫ Y ǫ t dt + AdW t . Then ∂ ∂ ǫ ln  ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T )  is b ounde d b elow, almost sur ely, uniformly for lar ge values of T . Pro of. L et ˙ Y ǫ t = ∂ ∂ ǫ Y ǫ t . Then we h a ve d ˙ Y ǫ t =  − Y ǫ t + F ǫ ˙ Y ǫ t  dt, or jointl y , d   Y ǫ t ˙ Y ǫ t   =   F ǫ 0 − I F ǫ     Y ǫ t ˙ Y ǫ t   dt +   A 0   dW t . 18 Since all eigen v alues of   F ǫ 0 − I F ǫ   ha ve negativ e real parts,   Y ǫ t ˙ Y ǫ t   is stable. Th erefore,               Y ǫ t ˙ Y ǫ t               = o ( t 1 4 + δ ) a . s . for some δ > 0 and 1 T Z T 0   Y ǫ t ˙ Y ǫ t    Y ǫ t ˙ Y ǫ t  dt is p ositiv e definite (since the RANK cond ition holds here as w ell) and it con verge s almost surely to some p ositiv e definite constant matrix as T → ∞ . Therefore, ( C ǫ T ) and ( ˙ C ǫ T ) ha ve the same order where C ǫ T = R T 0 Y ǫ t Y ǫ t dt and ˙ C ǫ T = R T 0 ˙ Y ǫ t ˙ Y ǫ t dt . Hence ( ˙ C ǫ T )( C ǫ T ) − 1 = O (1) a . s . as T → ∞ . (4.3) By Corollary 4.1, lim T →∞ ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) = 0 a . s . and lim T →∞ ( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ T ) = lim T →∞ ( ˙ Y ǫ T ) ′ ( ˙ C ǫ T ) − 1 ( ˙ Y ǫ T ) = 0 a . s . Consider ∂ ∂ ǫ ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) = 2( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 Y ǫ T + ( Y ǫ T ) ′ ∂ ∂ ǫ ( C ǫ T ) − 1 Y ǫ T = 2( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 Y ǫ T − ( Y ǫ T ) ′ ( C ǫ T ) − 1  ∂ ∂ ǫ C ǫ T  ( C ǫ T ) − 1 Y ǫ T ≥ − 2 h ( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ T ) i 1 / 2 h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i 1 / 2 − ( Y ǫ T ) ′ ( C ǫ T ) − 1 " Z T 0 ( Y ǫ u )( ˙ Y ǫ u ) ′ du + Z T 0 ( ˙ Y ǫ u )( Y ǫ u ) ′ du # ( C ǫ T ) − 1 Y ǫ T 19 ≥ − 2 h ( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ T ) i 1 / 2 h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i 1 / 2 − 2 h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i Z T 0 h ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) i 1 / 2 h ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) i 1 / 2 du ≥ − 2 h ( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ T ) i 1 / 2 h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i 1 / 2 − 2 h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i " Z T 0 ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) du + Z T 0 ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) du # = − 2 h ( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ T ) i 1 / 2 h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i 1 / 2 − 2 h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i . Therefore, ∂ ∂ ǫ ln h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i = h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i − 1 ∂ ∂ ǫ h ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) i ≥ − 2 " ( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ T ) ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) # 1 / 2 − 2 h p + T r h ( ˙ C ǫ T )( C ǫ T ) − 1 ii , whic h is b oun d ed b elo w (by a negativ e num b er p ossibly dep ending on ǫ ) uni- formly for large v alues of T by (4.3) and usin g the fact that b oth ( ˙ Y ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ T ) and ( ˙ Y ǫ T ) ′ ( ˙ C ǫ T ) − 1 ( ˙ Y ǫ T ) ha ve the same ord er and the latter has the order as that of ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ). Hence the pro of of Lemma 4.1. Pro of of Theorem 4.2. Let F ǫ = F − ǫI , ǫ > 0. Since all eigen v alues of F are on the left half sp ace, the real parts of all eigenv alues of F ǫ are negativ e, i.e., Y ǫ t is a stable p r o cess. By Corollary 4.1, lim T →∞ ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) = 0 . Let f ( ǫ ) = ln( Y ǫ T )( C ǫ T ) − 1 ( Y ǫ T ). Fix a n ǫ 1 > 0. f is a con tinuous function on [0 , ǫ 1 ] and is d ifferen tiable in (0 , ǫ 1 ). Then by the Mean V alue Th eorem, there exists an ǫ 0 ∈ (0 , ǫ 1 ) such that f ( ǫ 1 ) − f (0) = ǫ 1 ∂ ∂ ǫ f ( ǫ ) | ǫ = ǫ 0 . 20 That is, ( Y ǫ 1 T ) ′ ( C ǫ 1 T ) − 1 ( Y ǫ 1 T ) Y ′ T C − 1 T Y T ≥ exp  ǫ 1 ∂ ∂ ǫ f ( ǫ ) | ǫ = ǫ 0  , (4.4) whic h is uniformly p ositiv e (i.e., b ounded a wa y from zero) for large v alues of T by Lemma 4.1. S ince lim T →∞ ( Y ǫ T ) ′ ( C ǫ T ) − 1 ( Y ǫ T ) = 0 a . s . b y (4.4) lim T →∞ Y ′ T C − 1 T Y T = 0 a . s . Hence the p r o of of Theorem 4.2 . COROLLARY 4.2 With the same assumptions and notations as in L emma 4.1, || C − 1 / 2 T || = O ( T − 1 / 2 ) a . s . Pro of. C onsider ∂ ∂ ǫ T r[( C ǫ T ) − 1 ] = − 2T r " ( C ǫ T ) − 1 Z T 0 h ( Y ǫ u )( ˙ Y ǫ u ) ′ dt i ( C ǫ T ) − 1 # ≥ − 2T r( C ǫ T ) − 1 Z T 0 h ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) i 1 / 2 h ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) i 1 / 2 du ≥ − T r ( C ǫ T ) − 1 " Z T 0  ˙ Y ǫ u  ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) du + Z T 0 ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) du # = − T r( C ǫ T ) − 1  T r h ( C ǫ T ) − 1 ˙ C ǫ T i + T r h ( C ǫ T ) − 1 ( C ǫ T ) i . Hence ∂ ∂ ǫ ln T r[( C ǫ T ) − 1 ] ≥ −  T r h ( C ǫ T ) − 1 ˙ C ǫ T i + p  whic h is b ound ed b elo w (b y a negativ e n u m b er p ossibly dep ending on ǫ ) uniformly for large v alues of T . Therefore, as in (4.4), b y the Mean V alue T h eorem, T r[( C ǫ T ) − 1 ] T r[( C T ) − 1 ] is uniformly p ositiv e (i.e., b ounded aw a y fr om zero) for large v alues of T . Since 21 T r[( C ǫ T ) − 1 ] = O ( T − 1 ), w e ha ve O (T r[( C T ) − 1 ]) ≤ O (T r[( C ǫ T ) − 1 ]) = O ( T − 1 ). Again, as f or an y p ositiv e definite matrix K T , O ( || K T || ) = O (T r ( K T )), we obtain by Corollary 4.1(i), || ( C T ) − 1 / 2 || = || ( C ǫ T ) − 1 / 2 || = O ( T − 1 / 2 ). Hence the result. REMARK 4.2 It is clear f rom the argumen ts in th e ab o ve corollary 4.2 that, for th e eigen- v alues of F on the left h alf space, 1 T λ min ( C T ) = 1 T λ max ( C − 1 T ) > 0 , almost surely , uniformly for large v alues of T , since T λ max ( C − 1 T ) = T || C − 1 T || ≤ T O ( T − 1 ) = O (1) a . s . 5 General O r n stein-Uhlen b eck Pro cesses F or the Orns tein-Uhlen b ec k p ro cess defined in (1.1 ) with RANK condition (2.1), we ha ve considered the case in which all the eigenv alues of F h a ve p ositiv e real p arts and the case in which all the eige n v alues of F hav e zero or negativ e real parts (i.e., zero eigenv alues, pur ely imaginary and the eigen- v alues with negativ e real parts). No w w e com bin e these cases to discuss th e mixe d mo del in whic h F can b e decomp osed in to rational canonical form as follo ws: M F = GM =   G 0 0 0 G 1     M 0 M 1   =   G 0 M 0 G 1 M 1   , where all the charac teristic r o ots of G 0 lie in the righ t half space and all the c haracteristic ro ots of G 1 lie on the left half space. Let   U 0 t U 1 t   =   M 0 M 1   Y t = M Y t . 22 Then d   U 0 t U 1 t   = M d Y t = M F Y t dt + M AdW t =   G 0 0 0 G 1     U 0 t U 1 t   dt + M AdW t . Also, Z T 0 dW t Y ′ t ! M ′ =   R T 0 dW t U ′ 0 t R T 0 dW t U ′ 1 t   and M Z T 0 Y t Y ′ t dt ! M ′ =   R T 0 U 0 t U ′ 0 t dt R T 0 U 0 t U ′ 1 t dt R T 0 U 1 t U ′ 0 t dt R T 0 U 1 t U ′ 1 t dt   . Define, C 1 T = R T 0 U 1 t U ′ 1 t dt . W e no w deriv e the follo wing result. LEMMA 5.1 Supp ose, for the O rnstein-Uhlenb e c k pr o c ess define d in (1.1), the RANK c ondition (2.1) holds. Then Σ − 1 T = " D T M Z T 0 Y t Y ′ t dt ! M ′ D ′ T # − 1 →   B − 1 0 0 I p 1   a . s . (5.1) wher e B is define d in Se ction 2 (b ef or e (3.4)), I p 1 is a p 1 -dimensional iden- tity matrix and D T =   e − G 0 T 0 0 C − 1 / 2 1 T   . Pro of. O bserving (5.1), we obtain, by Theorem 3.1, that lim T →∞ e − G 0 T Z T 0 U 0 t U ′ 0 t dt ! e − G ′ 0 T = B is p ositiv e d efi nite a . s . Again, (Σ T ) 11 = C − 1 / 2 1 T C 1 T C − 1 / 2 1 T = I p 1 . Hence, the pro of is complete once w e sho w e − G 0 T ( R T 0 U 0 t U ′ 1 t dt ) C − 1 / 2 1 T → 0 p 0 × p 1 matrix almost surely , as T → ∞ . Notice that, by Corollary 3.1, lim T →∞ Z T 0 || e − G 0 T U 0 t || dt < ∞ a . s . 23 and from Th eorem 4.2 lim T →∞ U ′ 1 T C − 1 1 T U 1 T = 0 a . s . Therefore, for all ω outside a n ull set, and f or any giv en ǫ > 0, th ere exists T 0 ( ω ) > 0 su c h that for all t ≥ T 0 ( ω ), ( U ′ 1 t C − 1 1 t U 1 t ) 1 / 2 < ǫ/ (lim T →∞ R T 0 || e − G 0 T U 0 t ( ω ) || dt ). Hence           e − G 0 T ( Z T 0 U 0 t U ′ 1 t dt ) C − 1 / 2 1 T           ≤ Z T 0 || e − G 0 T U 0 t U ′ 1 t C − 1 / 2 1 T || dt ≤ Z T 0 ( ω ) 0 || e − G 0 T U 0 t || || C − 1 / 2 1 T U 1 t || dt + Z T T 0 ( ω ) || e − G 0 T U 0 t || || C − 1 / 2 1 T U 1 t || dt As T → ∞ , the first term go es to 0 since T 0 ( ω ) is fixed. The second term is less than ǫ by the choice of T 0 ( ω ) since C 1 t is increasing in t (in the sense that C 1 t 2 − C 1 t 1 is p ositiv e definite wh enev er t 2 > t 1 ) and || C − 1 / 2 1 T U 1 t || = ( U ′ 1 t C − 1 1 T U 1 t ) 1 / 2 ≤ ( U ′ 1 t C − 1 1 t U 1 t ) 1 / 2 . As ǫ is arbitrary , th e pro of is complete. W e no w observe that, ˆ F T − F = " T − 1 / 2 A Z T 0 dW t Y ′ t ! M ′ D ′ T # " D T M Z T 0 Y t Y ′ t dt ! M ′ D ′ T # − 1 × ( T 1 / 2 D T M ) and T − 1 / 2 A Z T 0 dW t Y ′ t ! M ′ D ′ T =   T − 1 / 2 e − G 0 T ( R T 0 U 0 t dW ′ t ) A ′ T − 1 / 2 C − 1 / 2 1 T ( R T 0 U 1 t dW ′ t ) A ′   ′ . The first term T − 1 / 2 A  R T 0 dW t U ′ 0 t  e − G ′ 0 T = O ( T − 1 / 2 ) b y Corollary 3.1(ii). T o s h o w th e remaining terms con verges to 0, we pro ve the follo win g The- orem. This theorem is in the spirit of Theorem 2.2 of W ei [25], whic h is present ed for the d iscrete case. 24 THEOREM 5.1 1 √ T Z T 0 dW t U ′ 1 t ! C − 1 / 2 1 T → 0 a . s . as T → ∞ . T o p ro ve Theorem 5.1, we need the follo w ing lemmas. LEMMA 5.2 Fix t 0 > 0 . Then, Z T t 0 U ′ 1 t C − 1 1 t U 1 t dt = O (log T ) a . s . as T → ∞ . Pro of. Notice that, d dt log | C 1 t | = T r  C − 1 1 t d dt C 1 t  = T r  C − 1 1 t U 1 t U ′ 1 t  = U ′ 1 t C − 1 1 t U 1 t , where | C 1 t | is the determinant of C 1 t . Ob serv e that, G 1 can b e further decomp osed into a rational canonical form as follo ws:      M 11 M 12 M 13      G 1 =      G 11 0 0 0 G 12 0 0 0 G 13           M 11 M 12 M 13      =      G 11 M 11 G 12 M 12 G 13 M 13      , where all th e c h aracteristic ro ots of G 11 ha ve negativ e real parts, those of G 12 are purely imaginary and th ose of G 13 are zero. F or i, j = 1 , 2 , 3, defi ne C 1 tij = R t 0 U 1 is U ′ 1 j s ds , where      U 11 s U 12 s U 13 s      =      M 11 M 12 M 13      U 1 s . Th us C 1 t = (( C 1 tij )) i,j =1 , 2 , 3 , and hence | C 1 t | ≤ | C 1 t 11 | | C 1 t 22 | | C 1 t 33 | . There- fore, by Th eorem 4.1 in S ection 4 and Theorems 7.1 and 7.2 in the App en dix, one obtains Z T t 0 U ′ 1 t C − 1 1 t U 1 t dt = log | C 1 T | | C 1 t 0 | = O (log T ) a . s . as T → ∞ . 25 Hence, the pr o of. W e observ e that, from Lemma 5.2, if w e let g ( T ) = R T t 0 U ′ 1 t C − 1 1 t U 1 t dt , then g ( T ) ↑ ∞ as T ↑ ∞ almost su rely . Also, E (log | C 1 T | ) = E ( P i log( λ i ( C 1 T ))) = P i E (log ( λ i ( C 1 T ))) ≤ P i log( E ( λ i ( C 1 T ))) ≤ p 1 log( E ( λ max ( C 1 T ))) ≤ p 1 log R T 0 E ( k U 1 t k 2 ) dt . It is clear that, for the eigen v alues on the left h alf space, E ( k U 1 t k 2 ) is at most O ( t k ), i.e., it gro ws at most lik e a p olynomial in t . Thus, E (log | C 1 T | ) = O (log T ) as well. Hence, using int egration by parts, w e obtain, E Z ∞ t 0 U ′ 1 t C − 1 1 t U 1 t t dt ! < ∞ . (5.2) LEMMA 5.3 L et M 1 T = R T 0 dW t U ′ 1 t . Then, u nder the hyp othesis of The or em 5.1, 1 T 1 / 2 M 1 T C − 1 / 2 T → 0 in probabilit y . Pro of. Notice that M 1 t is a martingale with resp ect to the filtration {F t } t ≥ 0 where F t = σ { W s : 0 ≤ s ≤ t } . Define N 1 T = R T t 1 dW t U ′ 1 t = M 1 T − M 1 t 1 . Then , for T > t 1 , N 1 T is al so a martingale. Define V t = T r[ C − 1 1 t M ′ 1 t M 1 t ] /t and ˜ V t = T r[ C − 1 1 t N ′ 1 t N 1 t ] /t . Since k 1 T 1 / 2 M 1 T C − 1 / 2 T k 2 ≤ V T ≤ 2 ˜ V T + 2T r ( C − 1 1 T M ′ 1 t 1 M 1 t 1 ] /T and T r( C − 1 1 T M ′ 1 t 1 M 1 t 1 ] /T → 0, almost surely , as T → ∞ , it is enough to show that ˜ V T → 0, in probabilit y , as T → ∞ and this would b e immediate once one sh o ws E ( ˜ V T ) → 0 as T → ∞ . No w use Itˆ o’s Lemma to get d ˜ V t = h T r  C − 1 1 t d ( N ′ 1 t N 1 t )  + T r h ( ˙ C − 1 1 t ) N ′ 1 t N 1 t i dt i t − ˜ V t t dt (5.3) where ˙ C − 1 1 t = − C − 1 1 t  ˙ C 1 t  C − 1 1 t = − C − 1 1 t U 1 t U ′ 1 t C − 1 1 t whic h is non -p ositiv e definite. Thus, 26 T r h ˙ C − 1 1 t  N ′ 1 t N 1 t i = − U ′ 1 t C − 1 1 t N ′ 1 t N 1 t C − 1 1 t U 1 t ≤ 0. Th erefore, b y (5.3) and applying the Itˆ o’s Lemma again, one obtains ˜ V T ≤ Z T t 1 T r  C − 1 1 t d ( N ′ 1 t N 1 t )  /t = Z T t 1 T r  C − 1 1 t  ( dN ′ 1 t ) N 1 t + N ′ 1 t ( dN 1 t ) + ( dN ′ 1 t )( dN 1 t )   /t. Define τ n = inf { t > t 1 : | ˜ V t | ≥ n } , then E ˜ V T ∧ τ n ≤ E Z T ∧ τ n t 1 T r  C − 1 1 t ( dN ′ 1 t )( dN 1 t )  /t (5.4) = E Z T ∧ τ n t 1 U ′ 1 t C − 1 1 t U 1 t t dt. Since V T ∧ τ n and U ′ 1 t C − 1 1 t U 1 t are n on-negativ e, by F atou’s Lemma and the Monotone Conv ergence Th eorem, E ˜ V T ≤ E Z T t 1 U ′ 1 t C − 1 1 t U 1 t (log t ) 1+ α dt. No w, b y the argumen t in (5.2), one has lim sup { T →∞} E ˜ V T ≤ αC t − α 1 . As t 1 can b e tak en to b e arbitrarily large, we ha ve the result. LEMMA 5.4 L et V t = T r[ C − 1 1 t M ′ 1 t M 1 t ] /t . Then, with the same assumptions and notations as in L emma 5.3 , Z ∞ t 1 E [ E ( dV t |F t )] + < ∞ . Pro of. Ap plying Itˆ o’s L emm a on V t , dV t =  T r h C − 1 1 t d ( M ′ 1 t M 1 t ) i + T r h ˙ C − 1 1 t ( M ′ 1 t M 1 t ) i dt  t − V t t dt where ˙ C − 1 1 t = − C − 1 1 t  ˙ C 1 t  C − 1 1 t = − C − 1 1 t U 1 t U ′ 1 t C − 1 1 t and T r h ˙ C − 1 1 t ( M ′ 1 t M 1 t ) i = − U ′ 1 t C − 1 1 t M ′ 1 t M 1 t C − 1 1 t U 1 t ≤ 0 . 27 Therefore, E ( dV t |F t ) ≤ E h T r( C − 1 1 t d ( M ′ 1 t M 1 t )) i /t | F t  = E h T r  C − 1 1 t [( dM ′ 1 t ) M 1 t + M ′ 1 t ( dM 1 t ) + ( dM ′ 1 t )( dM 1 t )] i /t | F t  = E h T r  C − 1 1 t ( dM ′ 1 t )( dM 1 t ) i /t | F t  = E U ′ 1 t C − 1 1 t U 1 t t dt | F t ! = U ′ 1 t C − 1 1 t U 1 t t dt. Th us, [ E ( dV t |F t )] + ≤ U ′ 1 t C − 1 1 t U 1 t t dt. Since U ′ 1 t C − 1 1 t U 1 t ≥ 0, by F ubini’s theorem and by (5.2) Z ∞ t 1 E [ E ( dV t |F t )] + = E Z ∞ t 1 [ E ( dV t |F t )] + ≤ E Z ∞ t 1 U ′ 1 t C − 1 1 t U 1 t t dt < ∞ . Hence, the pr o of. Pro of of Theorem 5.1. Define A t 1 ,T δ = { max t 1 δ } and H t 1 = { V t 1 ≤ ǫ } for an y ǫ > 0. Then, using the Lenglart Inequalit y (cf. Karatzas and Shr ev e [13] p30 or Lenglart [22]), P  A t 1 ,T δ ∩ H t 1  ≤ 1 δ E V t 1 I H t 1 + 1 δ Z T t 1 E  [ E ( dV t |F t )] + I H t 1  . Therefore, P  A t 1 ,T δ  = P  A t 1 ,T δ ∩ H c t 1  + P  A t 1 ,T δ ∩ H t 1  ≤ P  H c t 1  + P  A t 1 ,T δ ∩ H t 1  ≤ P  H c t 1  + 1 δ E V t 1 I H t 1 + 1 δ Z T t 1 E  [ E ( dV t |F t )] + I H t 1  ≤ P  H c t 1  + ǫ δ + 1 δ Z ∞ t 1 E [ E ( dV t |F t )] + , 28 whic h is finite since R ∞ t 1 E [ E ( dV t |F t )] + < ∞ b y Lemma 5.4. Therefore, as T → ∞ , P  lim T →∞ A t 1 ,T δ  = lim T →∞ P  A t 1 ,T δ  ≤ P  H c t 1  + ǫ δ + 1 δ Z ∞ t 1 E [ E ( dV t |F t )] + . Th us, lim su p t 1 →∞ P  lim T →∞ A t 1 ,T δ  ≤ ǫ δ . Since this is tr ue for all ǫ > 0, lim su p t 1 →∞ P  lim T →∞ A t 1 ,T δ  = 0 . This implies, 1 T 1 / 2 Z T 0 dW t U ′ 1 t ! C − 1 / 2 1 T → 0 a . s . Hence, the Th eorem. Pro of of T he orem 2.1. F rom Lemma 3.1, we hav e || e − G 0 T || = O ( e − λ 0 T ) and, from Corollary 4.2, we ha ve || C − 1 / 2 1 T || = O ( T − 1 / 2 ) almost sur ely , as T → ∞ . Thus, || T 1 / 2 D T M || = T 1 / 2 || M ||  || e − G 0 T || + || C − 1 / 2 1 T ||  = O (1) a . s . as T → ∞ . Therefore, from (5.1), C orollary 3.1(ii) and Theorem 5.1, w e ha v e lim T →∞ ˆ F T = F a.s. T o show that (2.3) holds, w e observe that, for the eigen v alues of F in the righ t half s p ace (2.3) follo ws from Theorem 3.1 and, f or the eigenv alues of F on the left h alf space (2.3) follo ws from arguments in Corollary 4.2 an d Remark 4.2. F or the mixed mo d el, w e observe Z T 0 Y t Y ′ t dt ! − 1 = D T M Σ T M ′ D ′ T 29 where lim T →∞ Σ T is a.s. p ositive definite. Thus, by Lemma 3.2, λ max   Z T 0 Y t Y ′ t dt ! − 1   = O ( λ max ( D T D ′ T )) = O ( T − 1 ) . Therefore, the Th eorem follo ws. 6 Asymptotic Efficiency In this section w e w ould like to sh o w that our estimator for the drift matrix F is asymp totically efficient even if the underlyin g pro cess is not n ecessarily stationary (stable). F or matrix-v alued estimator there sev eral w a ys to define asymptotic efficiency (see Barndorff-Nielson and S orensen [2], for details). The resu lt is already kno wn in one-dimensional case and for v ector-v alued parameters (e.g., [5, 7, 18, 23] and references therein) when the pro cesses are not necessarily stationary . F or m ulti-dimensional matrix-v alued case, similar things can b e pro ved once the asymptotic efficiency is prop erly d efi ned for the matrix v alued estimator. Observe that, when AA ′ is nonsin gular, the log -lik eliho o d of F , (see [5], pp. 21 3-214 ), on [0 , T ] is defined b y , L A ( F ) = R T 0 ( Y ′ t F ′ ( AA ′ ) − 1 d Y t ) − (1 / 2) R T 0 ( Y ′ t F ′ ( AA ′ ) − 1 F Y t ) dt. Thus, dL A ( F ) = tr " dF Z T 0 Y t d Y ′ t ! ( AA ′ ) − 1 − dF Z T 0 Y t Y ′ t dt ! F ′ ( AA ′ ) − 1 # . Therefore, dL A ( F ) /dF =  R T 0 d Y t Y ′ t   R T 0 Y t Y ′ t dt  − 1 . When AA ′ is not nonsingular, the log-lik eliho o d of F cannot b e written explicit ly . Therefore, M.L.E. of F could not b e ac hiev ed. Ho wev er, w e w ould sho w that the ab o ve estimator is asymp toti- cally efficien t u nder the assumptions of th e section 2. W e sho w that E (T r[( ˆ F T − F ) E ( C T )( ˆ F T − F ) ′ ]) 1 / 2 = O (1) as T → ∞ . 30 Let S T =  R T 0 AdW t Y ′ t  , and C T =  R T 0 Y t Y ′ t dt  as b efore. W e use T r[( ˆ F T − F ) E ( C T )( ˆ F T − F ) ′ ] = T r[ S T C − 1 T E ( C T ) C − 1 T S ′ T ] ≤ T r[ S T C − 1 T S ′ T ]T r[ C − 1 T E ( C T )] to prov e the follo wing result. Pro of of T he orem 2.2 Case 1: Eigen v alues of F are in the p ositiv e half space. Observe that, T r( S T C − 1 T S ′ T ) = T r( S T e − F ′ T ( e − F T C T e − F ′ T ) − 1 e − F T S ′ T ). S in ce S T e − F ′ T is a Gaussian pro- cess and its mean zero and v ariance e − F T E ( C T ) e − F ′ T con verge s (in fact, to E ( B )) as T → ∞ , S T e − F ′ T con verge s to a finite Gaussian random v ari- able in distribu tion. Also, from T heorem (3.1), as T → ∞ , e − F T C T e − F ′ T con verge s almost surely to B (wh ich is p ositive defin ite with probabilit y one). Thus, w e obtain T r( S T e − F ′ T ( e − F T C T e − F ′ T ) − 1 e − F T S ′ T ) con ve rges in distribution to fi nite random v ariable with finite exp ectation. No w, T r( C − 1 T E ( C T )) = T r(( e − F T C T e − F ′ T ) − 1 ( e − F T E ( C T ) e − F ′ T )), and from Theorem (3.1), as T → ∞ , ( e − F T C T e − F ′ T ) − 1 con verge s to B − 1 almost surely . Also, e − F T E ( C T ) e − F ′ T = R T 0 e − F t Y 0 Y ′ 0 e − F ′ t dt + R T 0 te − F t AA ′ e − F ′ t dt , whic h is fin ite as T → ∞ . Th us, it remains to s h o w, as T → ∞ , E ( e − F T C T e − F ′ T ) − 1 con verge s to E ( B − 1 ) (wh ic h is fi nite). First observe that, Z t − Y 0 = R t 0 e − F s AdW s is a symmetric (Gaussian) martingale and with E | Z t − Y 0 | 2 ≤ E | Z − Y 0 | 2 < ∞ . Th us M Z = max 0 ≤ t< ∞ ( Z t − Y 0 ) exists and has finite exp ectation. Also, (b y symmetry) m Z = min 0 ≤ t< ∞ ( Z t − Y 0 ) exists and has finite second mo- men t. F or symmetric matrices D 1 and D 2 , define, D 1 ≥ D 2 if D 1 − D 2 is non-negativ e definite. Therefore, e − F T C T e − F ′ T = Z T 0 e − F t Z T − t Z ′ T − t e − F ′ t dt ≥ Z T 0 e − F t ( m Z + Y 0 )( m Z + Y 0 ) ′ e − F ′ t dt 31 ≥ Z T 0 0 e − F t ( m Z + Y 0 )( m Z + Y 0 ) ′ e − F ′ t dt for all T ≥ T 0 , for s ome T 0 > 0 ( T 0 ma y b e tak en to b e 1). T h us, ( e − F T C T e − F ′ T ) − 1 ≤ ( R T 0 0 e − F t ( m Z + Y 0 )( m Z + Y 0 ) ′ e − F ′ t dt ) − 1 for all T ≥ T 0 . Since righ t h an d side has finite exp ectation, u sing dominated con- v ergence t yp e theorem dedu ce E ( B − 1 ) = lim T →∞ E ( e − F T C T e − F ′ T ) − 1 ≤ E ( R T 0 0 e − F t ( m Z + Y 0 )( m Z + Y 0 ) ′ e − F ′ t dt ) − 1 . Therefore, E (T r( C − 1 T E ( C T ))) is finite and hence E (T r[( ˆ F T − F ) E ( C T )( ˆ F T − F ) ′ ]) 1 / 2 = O (1). Case 2: Eigen v alues of F are on the left h alf space. When all the eigenv alues ha ve real parts negativ e, b y ergo dic theorem, lim T →∞ 1 T C T = R ∞ 0 e F t AA ′ e F ′ t dt = lim T →∞ E ( 1 T C T ). Thus, lim T →∞ E (T r ( S T C − 1 T S ′ T )) = lim T →∞ E (T r ( 1 T S ′ T S T ( R ∞ 0 e F t AA ′ e F ′ t dt ) − 1 )) = p , i.e., of O (1). Also, lim T →∞ E (T r ( C − 1 T E ( C T ))) = lim T →∞ E (T r (( 1 T C T ) − 1 E ( 1 T C T ))) = p . Th er efore, E (T r[( ˆ F T − F ) E ( C T )( ˆ F T − F ) ′ ]) 1 / 2 = O (1). Zero a nd purely imaginary eigenv alues. When the eigen v alues are either all purely imaginary or all zero, r eplace F b y F − ǫI = F ǫ , as it is done in Section 4, get the result as ab o v e b y ergod ic theorem. No w, as in Lemma 4.1, consider ∂ ∂ ǫ T r E (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) = 2T r E (( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 S ǫ T ) + T r E (( S ǫ T ) ′ ∂ ∂ ǫ ( C ǫ T ) − 1 S ǫ T ) = 2T r E (( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 S ǫ T ) − T r E (( S ǫ T ) ′ ( C ǫ T ) − 1  ∂ ∂ ǫ C ǫ T  ( C ǫ T ) − 1 S ǫ T ) ≥ − 2 E  h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2 h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2  32 − T r E " ( S ǫ T ) ′ ( C ǫ T ) − 1 " Z T 0 ( Y ǫ u )( ˙ Y ǫ u ) ′ du + Z T 0 ( ˙ Y ǫ u )( Y ǫ u ) ′ du # ( C ǫ T ) − 1 S ǫ T # ≥ − 2  E h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2  E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2 − 2 E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i Z T 0 h ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) i 1 / 2 h ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) i 1 / 2 du ! ≥ − 2  E h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2  E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2 − E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i " Z T 0 ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) du + Z T 0 ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) du #! = − 2  E h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2  E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2 − E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i . Therefore, ∂ ∂ ǫ ln E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i = h E T r (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i − 1 ∂ ∂ ǫ E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i ≥ − 2 " E T r (( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) E T r (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) # 1 / 2 − E   T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T ))  h p + T r h ( ˙ C ǫ T )( C ǫ T ) − 1 ii E [T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T ))] , whic h is b oun d ed b elo w (by a negativ e num b er p ossibly dep ending on ǫ ) uni- formly for large v alues of T by (4.3) and usin g the fact that b oth T r E (( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) and T r E (( ˙ S ǫ T ) ′ ( ˙ C ǫ T ) − 1 ( ˙ S ǫ T )) hav e the same ord er and the latter has the ord er as that of T r E (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )). No w as in the argument in consistency p art, sin ce all eigen v alues of F are on the left half space, the real parts of all ei gen v alues of F ǫ are negativ e, i.e., Y ǫ t is a stable p r o cess and lim T →∞ T r E (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) = O (1) . Similarly , to get a up p er b ound , consid er ∂ ∂ ǫ T r E (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) 33 = 2T r E (( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 S ǫ T ) − T r E (( S ǫ T ) ′ ( C ǫ T ) − 1  ∂ ∂ ǫ C ǫ T  ( C ǫ T ) − 1 S ǫ T ) ≤ 2 E  h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2 h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2  +T r E " ( S ǫ T ) ′ ( C ǫ T ) − 1 " Z T 0 ( Y ǫ u )( ˙ Y ǫ u ) ′ du + Z T 0 ( ˙ Y ǫ u )( Y ǫ u ) ′ du # ( C ǫ T ) − 1 S ǫ T # ≤ 2  E h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2  E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2 +2 E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i Z T 0 h ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) i 1 / 2 h ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) i 1 / 2 du ! ≤ 2  E h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2  E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2 + E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i " Z T 0 ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) du + Z T 0 ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) du #! = 2  E h T r(( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) i 1 / 2  E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i 1 / 2 + E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i . Therefore, ∂ ∂ ǫ ln E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i = h E T r (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i − 1 ∂ ∂ ǫ E h T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) i ≤ 2 " E T r (( ˙ S ǫ T ) ′ ( C ǫ T ) − 1 ( ˙ S ǫ T )) E T r (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) # 1 / 2 + E   T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T ))  h p + T r h ( ˙ C ǫ T )( C ǫ T ) − 1 ii E [T r(( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T ))] , whic h is b ounded ab ov e (b y a p ositiv e num b er p ossibly dep ending on ǫ ) uniformly for large v alues of T b y (4.3). As in the pr o of of T heorem 4.2, let f ( ǫ ) = ln T r E (( S ǫ T )( C ǫ T ) − 1 ( S ǫ T )). Fix an ǫ 1 > 0. f is a con tin uou s fu n ction on [0 , ǫ 1 ] and is d ifferentiable in (0 , ǫ 1 ). Then by the Mean V alue T heorem, there exists an ǫ 0 ∈ (0 , ǫ 1 ) such that f ( ǫ 1 ) − f (0) = ǫ 1 ∂ ∂ ǫ f ( ǫ ) | ǫ = ǫ 0 . 34 That is, T r E (( S ǫ 1 T ) ′ ( C ǫ 1 T ) − 1 ( S ǫ 1 T )) T r E ( S ′ T C − 1 T S T ) = exp  ǫ 1 ∂ ∂ ǫ f ( ǫ ) | ǫ = ǫ 0  , (6.1) whic h is uniformly b ounded and p ositiv e (i.e., b ou n ded a wa y from zero and infinity) for large v alues of T as argued ab ov e. Since lim T →∞ T r E (( S ǫ T ) ′ ( C ǫ T ) − 1 ( S ǫ T )) = O (1) . b y (6.1) lim T →∞ T r E ( S ′ T C − 1 T S T ) = O (1) . Mimic king the ab ov e argument, find ∂ ∂ ǫ T r  E (( C ǫ T ) − 1 ) E ( C ǫ T )  = − T r E ( C ǫ T ) − 1 " Z T 0 ( Y ǫ u )( ˙ Y ǫ u ) ′ du + Z T 0 ( ˙ Y ǫ u )( Y ǫ u ) ′ du # ( C ǫ T ) − 1 ! E ( C ǫ T ) ! +T r E (( C ǫ T ) − 1 ) E " Z T 0 ( Y ǫ u )( ˙ Y ǫ u ) ′ du + Z T 0 ( ˙ Y ǫ u )( Y ǫ u ) ′ du #! ≥ − 2 E h T r(( C ǫ T ) − 1 E ( C ǫ T )) i Z T 0 h ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) i 1 / 2 h ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) i 1 / 2 du ! − 2 E Z T 0 h ( Y ǫ u ) ′ ( E (( C ǫ T ) − 1 ))( Y ǫ u ) i 1 / 2 h ( ˙ Y ǫ u ) ′ ( E (( C ǫ T ) − 1 ))( ˙ Y ǫ u ) i 1 / 2 du ! ≥ − E h T r(( C ǫ T ) − 1 E ( C ǫ T )) i h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i − 2 E  h T r( E (( C ǫ T ) − 1 ))( C ǫ T ) i 1 / 2 h T r( E (( C ǫ T ) − 1 ))( ˙ C ǫ T ) i 1 / 2  ≥ − E h T r(( C ǫ T ) − 1 E ( C ǫ T )) i h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i − 2  T r h E (( C ǫ T ) − 1 ) E ( C ǫ T ) i 1 / 2  T r h E (( C ǫ T ) − 1 ) E ( ˙ C ǫ T ) i 1 / 2 Therefore, ∂ ∂ ǫ ln T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) 35 = h T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) i − 1 ∂ ∂ ǫ T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) ≥ − E   T r(( C ǫ T ) − 1 E ( C ǫ T ))  h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) − 2 " T r( E (( C ǫ T ) − 1 ) E ( ˙ C ǫ T )) T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) # 1 / 2 , whic h is b oun d ed b elo w (by a negativ e num b er p ossibly dep ending on ǫ ) uni- formly for large v alues of T by (4.3) and usin g the fact that b oth T r( E (( C ǫ T ) − 1 ) E ( ˙ C ǫ T )) and T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) ha v e the same order. Similary , to get an up p er b oun d, consider ∂ ∂ ǫ T r  E (( C ǫ T ) − 1 ) E ( C ǫ T )  = − T r E ( C ǫ T ) − 1 " Z T 0 ( Y ǫ u )( ˙ Y ǫ u ) ′ du + Z T 0 ( ˙ Y ǫ u )( Y ǫ u ) ′ du # ( C ǫ T ) − 1 ! E ( C ǫ T ) ! +T r E (( C ǫ T ) − 1 ) E " Z T 0 ( Y ǫ u )( ˙ Y ǫ u ) ′ du + Z T 0 ( ˙ Y ǫ u )( Y ǫ u ) ′ du #! ≤ 2 E h T r(( C ǫ T ) − 1 E ( C ǫ T )) i Z T 0 h ( Y ǫ u ) ′ ( C ǫ T ) − 1 ( Y ǫ u ) i 1 / 2 h ( ˙ Y ǫ u ) ′ ( C ǫ T ) − 1 ( ˙ Y ǫ u ) i 1 / 2 du ! +2 E Z T 0 h ( Y ǫ u ) ′ ( E (( C ǫ T ) − 1 ))( Y ǫ u ) i 1 / 2 h ( ˙ Y ǫ u ) ′ ( E (( C ǫ T ) − 1 ))( ˙ Y ǫ u ) i 1 / 2 du ! ≤ E h T r(( C ǫ T ) − 1 E ( C ǫ T )) i h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i +2 E  h T r( E (( C ǫ T ) − 1 ))( C ǫ T ) i 1 / 2 h T r( E (( C ǫ T ) − 1 ))( ˙ C ǫ T ) i 1 / 2  ≤ E h T r(( C ǫ T ) − 1 E ( C ǫ T )) i h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i +2  T r h E (( C ǫ T ) − 1 ) E ( C ǫ T ) i 1 / 2  T r h E (( C ǫ T ) − 1 ) E ( ˙ C ǫ T ) i 1 / 2 Therefore, ∂ ∂ ǫ ln T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) = h T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) i − 1 ∂ ∂ ǫ T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) ≤ E   T r(( C ǫ T ) − 1 E ( C ǫ T ))  h p + T r[( ˙ C ǫ T )( C ǫ T ) − 1 ] i T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) + 2 " T r( E (( C ǫ T ) − 1 ) E ( ˙ C ǫ T )) T r( E (( C ǫ T ) − 1 ) E ( C ǫ T )) # 1 / 2 , 36 whic h is b ounded ab ov e (b y a p ositiv e num b er p ossibly dep ending on ǫ ) uniformly for large v alues of T b y (4.3). Th us, u sing the similar argumen t as in (6.1) we sho w, since lim T →∞ T r( E (( C ǫ 1 T ) − 1 ) E ( C ǫ 1 T )) = O (1), lim T →∞ T r( E ( C − 1 T ) E ( C T )) = O (1) . Hence, for eigen v alues of F on the left half space, w e prov e that E (T r[( ˆ F T − F ) E ( C T )( ˆ F T − F ) ′ ]) 1 / 2 = O (1). Case 3: Mixed mo d el. In this case, use the decomp osition of F as in S ection 5, to decomp ose Y ′ t M ′ = ( U ′ 0 t , U ′ 1 t ). Then, one gets, tr ( S T C T − 1 S ′ T ) = tr ( S T M ′ D ′ T ( D T M C T M ′ D ′ T ) − 1 D T M S ′ T ) ≤ tr ( S T M ′ D ′ T D T M S ′ T ) tr ( D T M C T M ′ D ′ T ) − 1 ≤ ( tr ( S 0 T e − G ′ 0 T e − G 0 T S ′ 0 T ) + tr ( S 1 T C 1 T − 1 S ′ 1 T )) tr ( D T M C T M ′ D ′ T ) − 1 . Since for a symmetric in vertible partition matrix, K =   E F F ′ H   with E and H in vertible, tr ( K ) = tr ( E − F H − 1 F ′ ) − 1 + tr ( H − F ′ E − 1 F ) − 1 . T aking E = e − G 0 T C 0 T e − G ′ 0 T , F = e − G 0 T R T 0 U 0 t U ′ 1 t dtC − 1 / 2 1 T and H = I , i.e., iden tit y matrix of order p 1 . Since F conv erging to zero almost surely b y the p ro of of Lemm a 5.1 and b y the same lemma E conv er ges to B almost surely , one obtains tr ( D T M C T M ′ D ′ T ) − 1 → tr ( B − 1 ) + p 1 almost sur ely , as T → ∞ . Therefore, E h ( tr ( e − G 0 T S ′ 0 T S 0 T e − G ′ 0 T ) + tr ( S 1 T C 1 T − 1 S ′ 1 T )) tr ( D T M C T M ′ D ′ T ) − 1 i 1 / 2 ≤ E ( tr ( e − G 0 T S ′ 0 T S 0 T e − G ′ 0 T )) E ( tr ( D T M C T M ′ D ′ T ) − 1 ) + E ( tr ( S 1 T C 1 T − 1 S ′ 1 T )) E ( tr ( D T M C T M ′ D ′ T ) − 1 ) = O (1) (6.2) 37 b y the case 1, and case 2. Similarly , tr (( D T M C T M ′ D ′ T ) − 1 D T E ( M C T M ′ ) D ′ T ) ≤ tr (( D T M C T M ′ D ′ T ) − 1 ) tr ( D T E ( M C T M ′ ) D ′ T ) and tr ( D T E ( M C T M ′ ) D ′ T ) = tr ( e − G 0 T E ( C 0 T ) e − G ′ 0 T ) + tr ( C − 1 1 T E ( C 1 T )) ex- p ectation of whic h is fin ite b y case 1 and case 2. Therefore one pro ve s, for the mixed mo del, E (T r[( ˆ F T − F ) E ( C T )( ˆ F T − F ) ′ ]) 1 / 2 = O (1). Concluding remarks and discussion It is easy to see that the state space equation of the ge neral con tin uous autoregressiv e p ro cess (CAR(p)) of the form dX p − 1 t = α p X t + α p − 1 X 1 t + · · · + α 1 X p − 1 t + σ dW t is a sp ecial case of m u ltidimensional OU pro cesses where F =   0 ( p − 1) × 1 I p − 1 α p · · · α 1   , A = (0 , · · · , 0 , σ ) ′ with α i real n umber s , σ > 0 and W t a one-dimensional Bro w ian motion. Clearly , A is n ot singular. How ev er, the RANK condition (a) holds for this F and A and, the condition (b’) h olds for this F . Hence, from our result, the consistency and the asymptotic efficiency of the ˆ F of general CAR(p) follo ws. It is imp ortant to observ e that this estimation pro cedure ma y b e the firs t step in deve loping a test of zero ro ots of some F , whic h is necessary to de- termine whether u niv ariate pro cesses are co-in tegrated. Also, if one needs to dev elop a test to d etermin e wh ether the mo del for Y t is stationary , it is often enough to test whether all eigen v alues of F ha ve negativ e real parts against the alternativ e that some of them h a ve zero real parts. Th er e- fore, one need not often worry ab ou t the assump tion (b) or (b’) for testing stationarit y . Thus, a related question arises on, whether an y Asy m ptoti- cally Mixed Normalit y p rop erty holds for the estimato r ˆ F T , i.e., whether 38 ( R T 0 Y t Y ′ t dt ) 1 / 2 ( ˆ F T − F ) follo ws asymptotically Normal, so that w e could compute approximate confid ence inte rv al for the ab ov e testing pro cedures for the necessary parameters in F . As far as w e kn o w, these r esu lts are still u n kno wn. Inv estigating the Asymptotically Mixed Normalit y prop ert y ma y b e an imp ortan t futur e dir ection to consider. One can look in to LAMN prop erty as w ell. Besides, wh en the d r ift co efficien t matrix dep end s on an unkn o wn discrete paratmeter θ whic h follo ws a Mark ov chain (that h elps the pro cess to sw itc h regimes), find ing a consisten t and asymptotically efficien t estimator b ecomes imp ortant. Ab o ve questions can b e ask ed in that setup as w ell. In applications, we almost alwa ys use discrete sampled data. Similar qu es- tions can b e ask ed for this mo del, w hen the d ata sampled are in d etermin istic (equal or unequal) time interv al or in random in terv al. T h at can also b e a fo cus of the f u ture direction. 7 App endix 7.1 Purely Imaginary Eigen v alues In this Section, we study the asymp totic b eha vior of OU processes when the drift matrix F only con tains purely imaginary eig en v alues. Th e main results are su mmarized in the follo w in g: THEOREM 7.1 Supp ose for the Ornstein-Uhlenb e ck pr o c e ss define d in (1.1), the RANK c ondition (2.1) holds and al l the eigenvalues of F a r e pur ely imag- inary. L et 2 ρ b e the dimension of the lar gest blo ck of the r ational c anonic al 39 form of F as define d in Se ction 2 (se e the Example). Then || Y T || =    O ( T 1 / 2 √ ln ln T ) a . s . if ρ = 1 O ( T 2 ρ − 5 / 2 √ ln ln T ) a . s . if ρ ≥ 2 . Mor e over, λ max Z T 0 Y t Y ′ t dt ! =    O ( T 2 (ln ln T )) a . s . if ρ = 1 O ( T 4 ρ − 4 (ln ln T )) a . s . if ρ ≥ 2 . (7.1) T o p ro ve Theorem 7.1, we need the follo w ing Lemmas. LEMMA 7.1 ∞ X n = j ( − 1) n ( v t ) 2 n − j (2 n − j )! =    O (1) if j = 0 , 1 O ( t j − 2 ) if j ≥ 2 . Pro of. ∞ X n = j ( − 1) n ( v t ) 2 n − j (2 n − j )! = ( − 1) j " ( v t ) j j ! − ( v t ) j +2 ( j + 2)! + ( v t ) j +4 ( j + 4)! − · · · # =                cos( v t ) if j = 0 − sin( vt ) if j = 1 ( − 1) j / 2 n cos( v t ) − h 1 − ( vt ) 2 2! + · · · + ( − 1) j / 2 − 1 ( vt ) j − 2 ( j − 2)! io if j is ev en , j ≥ 2 ( − 1) ( j − 3) / 2 n sin( v t ) − h v t − ( vt ) 3 3! + · · · + ( − 1) ( j − 1) / 2 ( vt ) j − 2 ( j − 2)! io if j is od d , j ≥ 3 =    O (1) if j = 0 , 1 O ( t j − 2 ) if j ≥ 2 . Hence, the lemma follo w s . LEMMA 7.2 With the same assumptions as in The or em 7.1, || e F t || =    O (1) a . s . if ρ = 1 O ( t 2 ρ − 3 ) a . s . if ρ ≥ 2 . . 40 Pro of. Supp ose F is a 2 ρ × 2 ρ mat rix and has ρ eigen v alues of λ 1 = iv and ¯ λ 1 = − iv . Since the c h aracteristic equatio n for F is 0 = | λI − F | = ( λ − iv ) ρ ( λ + iv ) ρ = ( λ 2 + v 2 ) ρ , by the Ca yley-Hamilton theorem, ( F 2 + v 2 I ) ρ = 0 . (7.2) Case 1 : When ρ = 1, then F 2 n = ( − 1) n v 2 n I and e F t = ∞ X n =0 F 2 n t 2 n (2 n )! + F ∞ X n =0 F 2 n t 2 n +1 (2 n + 1)! = I ∞ X n =0 ( − 1) n ( v t ) 2 n (2 n )! + F v ∞ X n =0 ( − 1) n ( v t ) 2 n +1 (2 n + 1)! = I cos( v t ) + F v sin( v t ) . (7.3) Therefore, || e F t || = O (1) wh en ρ = 1. Case 2 : When ρ ≥ 2, then A = F 2 + v 2 I is a nilp oten t matrix of order ρ b y (7.2). Thus, F 2 = − v 2  I − A v 2  and F 2 n = ( − 1) n v 2 n ρ − 1 X k =0 ( − 1) k n k ! A k v 2 k = ( − 1) n v 2 n I − nA v 2 + · · · + ( − 1) ρ − 1 n ρ − 1 ! A ρ − 1 v 2( ρ − 1) ! . Therefore, e F t = ∞ X n =0 F 2 n t 2 n (2 n )! + F ∞ X n =0 F 2 n t 2 n +1 (2 n + 1)! . (7.4) Let f j ( n ) = 2 n (2 n − 1) · · · (2 n − j + 1) if j ≥ 1 and f 0 ( n ) = 1. Then, since f 0 ( n ) , f 1 ( n ) , · · · , f k ( n ) are indep endent , there exist u nique C 0 , C 1 · · · C k ∈ Z suc h that n k ! = k X j =0 C j f j ( n ) . 41 Similarly , let f ∗ j ( n ) = (2 n + 1)(2 n ) · · · (2 n − j + 2) if j ≥ 1 and f ∗ 0 ( n ) = 1. Then, there exist unique C ∗ 0 , C ∗ 1 , · · · C ∗ k ∈ Z suc h that n k ! = k X j =0 C ∗ j f ∗ j ( n ) . By Lemma 7.1, th e first term of (7.4 ) can b e expressed as ∞ X n =0 ( − 1) n ( v t ) 2 n (2 n !)   ( ρ − 1) ∧ n X k =0 ( − 1) k n k ! A k v 2 k   = ρ − 1 X k =0  − A v 2  k   ∞ X n = k ( − 1) n ( v t ) 2 n (2 n )!   k X j =0 C j f j ( n )     = ρ − 1 X k =0  − A v 2  k   k X j =0 ( v t ) j C j ∞ X n = k ( − 1) n ( v t ) 2 n − j (2 n − j )! !   =          1 P k =0  − A v 2  k × O ( t ) f or ρ = 2 ρ − 1 P k =0  − A v 2  k × O ( t 2 k − 2 ) for ρ ≥ 3 =    O ( t ) for ρ = 2 O ( t 2 ρ − 4 ) for ρ ≥ 3 . Similarly , the second term of (7.4) can b e expressed as F ∞ X n =0 F 2 n t 2 n +1 (2 n + 1)! = F v ∞ X n =0 ( − 1) n ( v t ) 2 n +1 (2 n + 1)! ( ρ − 1) ∧ n X k =0 ( − 1) k n k ! A k v 2 k = F v ρ − 1 X k =0  − A v 2  k   k X j =0 ( v t ) j C j ∞ X n = k ( − 1) n ( v t ) 2 n − j +1 (2 n − j + 1)! !   = F v ρ − 1 X k =0  − A v 2  k × O ( t 2 k − 1 ) = O ( t 2 ρ − 3 ) . 42 Hence, the Lemma follo ws. LEMMA 7.3 Z T 0 ( T − s ) k AdW s = O  T k +1 / 2 √ ln ln T  Pro of. Let M u = R u 0 ( t − s ) k AdW s , which is a square inte grable martin- gale for [0 < u ≤ t ] and < M > u = R u 0 ( t − s ) 2 k AA ′ ds = [ t 2 k + 1 − ( t − u ) 2 k + 1 ] AA ′ / (2 k + 1). Since M u = B u b y Karatzas and Shr ev e ([13] p174), Z T 0 ( T − s ) k AdW s = O ( B T 2 k +1 ) = O ( T k +1 / 2 √ ln ln T ) . Hence, the lemma follo w s . Pro of of T heorem 7.1. If ρ = 1, then there exist C ∈ R suc h that || e F t || ≤ C by (7.3). Therefore, || Y T || = || e F T Y 0 + Z T 0 e F ( T − s ) AdW s || ≤ C Y 0 + C h O ( √ T ln ln T ) i = O ( √ T ln ln T ) . F or ρ ≥ 2, b y Lemm a 7.2 and 7.3, || Y T || = || e F T Y 0 + Z T 0 e F ( T − s ) AdW s || ≤ O   || e F T Y 0 || + || Z T 0 2 ρ − 3 X k =0 C k ( T − s ) k AdW s ||   = O   || e F T Y 0 || + || 2 ρ − 3 X k =0 C k Z T 0 ( T − s ) k AdW s ||   ≤ O   || e F T Y 0 || + 2 ρ − 3 X k =0 | C k | × || O ( T k +1 / 2 √ ln ln T ) ||   = O ( T 2 ρ − 5 / 2 √ ln ln T ) . T o s ho w (7.1), w e hav e λ max Z T 0 Y t Y ′ t dt ! = O tr Z T 0 Y t Y ′ t dt ! 43 = O Z T 0 || Y t || 2 dt ! =    O ( T 2 (ln ln T )) a . s . if ρ = 1 O ( T 4 ρ − 4 (ln ln T )) a . s . if ρ ≥ 2 . Hence, the pr o of of the theorem. 7.2 Zero E igen v alues In this Section, we study the asymptotic b eha vior of the OU pro cesses w hen the dr ift matrix F con tains only zeros eigen v alues.(i.e., F is a n ilp oten t matrix.) Th e main results are s ummarized in the follo wing: THEOREM 7.2 Supp ose for the OU pr o c ess define d in (1.1), the RANK c ondition (2.1) holds and, al l eigenvalues of F ar e zer os. L et γ b e the di- mension of the lar gest blo ck of the r ational c anonic al form of F as define d in Se c tion 2 (i.e., F γ = 0 ; se e the Example). Then || Y T || = O ( T γ − 1 / 2 √ ln ln T ) a . s . Mor e over, λ max Z T 0 Y t Y ′ t dt ! = O ( T 2 γ (ln ln T )) a . s . (7.5) Pro of. Since F is a k × k nilp oten t matrix of order γ (1 ≤ γ ≤ k ), th en F γ = 0 and e F t = γ − 1 X n =0 F n t n n ! = O ( t γ − 1 ) . || Y T || ≤ O   || e F T Y 0 || + Z T 0 γ − 1 X k =0 C k ( T − s ) k AdW s   44 = O ( || e F T Y 0 || ) + O   γ − 1 X k =0 C k Z T 0 ( T − s ) k AdW s   = O ( T γ − 1 ) + O ( T γ − 1 / 2 √ ln ln T ) = O ( T γ − 1 / 2 √ ln ln T ) . T o p ro ve (7.5) λ max Z T 0 Y t Y ′ t dt ! = O T r Z T 0 Y t Y ′ t dt ! = O Z T 0 || Y t || 2 dt ! = O ( T 2 γ (ln ln T )) . Hence, the pr o of. 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