1. Introduction. Tis paper is concerned wit estimation of te parameters of a non-negative Levydriven Ornstein-Ulenbeck process and of te parameters of

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1 Estimation for on-negative Levy-driven Ornstein-Ulenbeck Processes Peter J. Brockwell, Ricard A. Davis and Yu Yang, Department of Statistics, Colorado State University, Fort Collins, CO Abstract Continuous-time autoregressive moving average (CARMA) processes wit a nonnegative kernel and driven by a non-decreasing Levy process constitute a very general class of stationary, non-negative continuous-time processes. An example is te Levy-driven stationary Ornstein-Ulenbeck (or CAR(1)) process, introduced by Barndor-ielsen and Separd (21) as a model for stocastic volatility. For suc processes we take advantage of te non-negativity ofte increments of te driving Levy process to develop a igly ecient estimation procedure for te parameters wen observations are available at uniformly spaced times :::. We also sow ow to reconstruct te background driving Levy process from a continuously observed realization of te process and use tis result to estimate te increments of te Levy process itself wen is small. Asymptotic properties of te coecient estimator are derived and te results illustrated using a simulated gamma-driven Ornstein-Ulenbeck process. AMS 2 Matematics Subject Classication : Primary : 62M1, 6H1. Secondary: 62M9. Keywords and prases: Continuous-time autoregression, Ornstein-Ulenbeck process, Levy process, stocastic dierential equation, sampled process. 1

2 1. Introduction. Tis paper is concerned wit estimation of te parameters of a non-negative Levydriven Ornstein-Ulenbeck process and of te parameters of te background driving Levy process, based on observations made at uniformly and closely-spaced times. Te idea is to obtain a igly ecient estimator of te CAR(1) coecient by estimating te corresponding coecient of te sampled AR(1) process using te estimator of Davis and McCormick (1989) for non-negative discrete-time AR(1) processes. Tis estimator is ten used to estimate te corresponding realization of te driving Levy process using a generalization of an argument due to Pam-Din-Tuan (1977). Te exact recovery of te driving Levy process requires continuous observation of te Ornstein-Ulenbeck process. Te integral expressions determining te driving Levy process are terefore replaced by approximating sums using te available discrete-time observations. In Section 2, we dene te stationary Levy-driven Ornstein-Ulenbeck (or CAR(1)) process, fy (t) t g. In Section 3, we caracterize te sampled AR(1) process, fy n = Y (n) n = 1 2 :::g, and te distribution of its driving wite noise sequence in terms of te parameters of te underlying CAR(1) process and its driving Levy process. Te autoregressive coecient of te sampled process is ten estimated wit very ig eciency using te metod of Davis and McCormick (1989). From te relation between te sampled and continuous-time processes we ten obtain corresponding parameter estimates for te CAR(1) process. Te idea of using te sampled process to estimate te parameters of te underlying continuous-time process was rst used by Pillips (1954), but in our case te non-decreasing property of te driving Levy process and te non-negativity of te corresponding discrete-time increments permits a very large eciency gain. In Section 4, we sow ow to recover te driving Levy process under te assumption tat te process is observed continuously and ten approximate te results using closely-spaced discrete observations. In Section 5, we derive te asymptotic distribution of te coecient estimator wen te driving Levy process is a gamma process and illustrate wit a simulated example te performance of te estimators of bot te CAR(1) parameters and te driving Levy process. Wen te continuously observed process is available, te autoregression coecient can be identied wit probability 1.Tis is discussed in Section Stationary Levy-driven Ornstein-Ulenbeck processes. In order to dene te stationary Levy-driven Ornstein Ulenbeck (or CAR(1)) process, we rst record a few essential facts concerning Levy processes. (For a detailed account of integration wit respect to Levy processes, see Protter (24)). Suppose we are given a 2

3 ltered probability space ( F (F t ) t1 P), were F contains all te P -null sets of F and (F t )isrigt-continuous. Denition 1 (Levy Process). fl(t) t g is an (F t )-adapted Levy process if L(t) 2F t for all t and L() = a.s., L(t) ; L(s) is independent of F s, s<t<1, L(t) ; L(s) as te same distribution as L(t ; s) and L(t) is continuous in probability. Every Levy process as a unique modication wic is cadlag (rigt continuous wit left limits) and wic is also a Levy process. We sall terefore assume tat our Levy process as tese properties. For non-decreasing Levy processes te Laplace transform ~f L(t) (s) :=E(exp(;sL(t))) as te form were ~f L(t) (s) = exp(;t(s)) <(s) (s) =m + Z ( 1) (1 ; e ;sx )(dx) wit m and a measure on te Borel subsets of ( 1) satisfying Z ( 1) u (du) < 1: 1+u Te measure is called te Levy measure of te process L and m is te drift. Tere exists a wealt of possible marginal distributions for L(t), attainable by suitable coice of m and. (See for example Barndor-ielsen and Separd (21).) For second-order Levy processes E(L(1)) 2 < 1 and tere exist real constants and suc tat EL(t) =t and Var(L(t)) = 2 t for t : To avoid problems of parameter identiability in te CAR(1) process dened below we assume trougout tat L is scaled so tat Var(L(1)) = 1. Ten Var(L(t)) = t for t and we sall refer to te process L as a standardized second-order Levy process. Trougout tis paper we sall be concerned wit CAR(1) (or stationary Ornstein-Ulenbeck) processes driven by standardized second-order non-decreasing Levy processes. Te Levydriven CAR(1) process is dened as follows. 3

4 Denition 2 (Levy-driven CAR(1) process) A CAR(1) process driven by te Levy process fl(t) t g is dened to be a strictly stationary solution of te stocastic dierential equation, dy (t)+ay (t)dt = dl(t): (2:1) In te special case wen fl(t)g is Brownian motion, (2.1) is interpreted as an It^o equation wit solution fy (t) t g satisfying Y (t) =e ;at Y () + e ;a(t;u) dl(u) (2:2) were te integral is dened as te L 2 limit of approximating Riemann-Stieltjes sums. For any second-order driving Levy process, fl(t)g, te process fy (t)g can be dened in te same way, and if fl(t)g is non-decreasing (and ence of bounded variation on compact intervals) fy (t)g can also be dened patwise as a Riemann-Stieltjes integral by (2.2). We can also write Y (t) =e ;a(t;s) Y (s)+ s e ;a(t;u) dl(u) for all t>s (2:3) sowing, by independence of te increments of fl(t)g, tat fy (t)g is Markov. (For a general teory of integration wit respect to semimartingales, and in particular wit respect to Levy processes see Protter (24).) Te following proposition gives necessary and sucient conditions for stationarity of fy (t)g. For a proof see Brockwell and Marquardt (25). Proposition 1. If Y () is independent of fl(t) t g and E(L(1) 2 ) < 1, ten Y (t) is strictly stationary if and only if a> and Y () as te distribution of R 1 e ;au dl(u). Remark 1. By introducing a second Levy process fm(t) t<1g, independent of L and wit te same distribution, we can extend fy (t) t g to a process wit index set (;1 1). Dene te following extension of L: L (t) =L(t)I [ 1) (t) ; M(;t;)I (;1 ] (t) ;1<t<1: Ten, provided a>, te process fy (t)g dened by Y (t) = ;1 e ;a(t;u) dl (u) (2:4) is a strictly stationary process satisfying equation (2.3) (wit L replaced by L ) for all t>sand s 2 (;1 1). Hencefort, we refer to L as te background driving Levy process (BDLP) and denote it by L for simplicity. 4

5 Remark 2. From (2.4) we ave te relation Y (t) =e ;a(t;s) Y (s)+ s e ;a(t;u) dl(u) t s>;1: (2:5) Taking s =and using Lemma 2.1 of Eberlein and Raible (1999), we nd tat Y (t) =e ;at Y () + L(t) ; a e ;a(t;u) L(u)du t (2:6) were te last integral is a Riemann integral and te equality olds for all nite t wit probability Parameter estimation via te sampled process. Setting t = n and s =(n ; 1) in equation (2.5), we see at once tat for any >, te sampled process fy n n= 1 2 :::g is te discrete-time AR(1) process satisfying Y n = Y n;1 + Z n n = 1 2 ::: (3:1) were = e ;a (3:2) and Z n = Z n (n;1) e ;a(n;u) dl(u): (3:3) Te noise sequence fz n g is i.i.d. and positive since L as stationary, independent and positive increments. If te process fy (t) t T g is observed at times 2 :::, were = [T=], i.e., is te integer part of T=, ten, since te innovations Z n of te process fy n g are non-negative and <<1, we canusete igly ecient Davis-McCormick estimator of, namely ^ = min 1n Y n Y n;1 : (3:4) To obtain te asymptotic distribution of ^ as!1wit xed, we need to suppose tat te distribution function F of Z n satises F () = and tat F is regularly varying at zero wit exponent, i.e., tat tere exists >suc tat lim t# F (tx) F (t) = x for all x>: 5

6 (Tese conditions are satised by te gamma-driven CAR(1) process as we sall sow in Section 5.) Under tese conditions on F, te results of Davis and McCormick (1989) imply tat ^! a:s: as!1wit xed and tat lim P!1 k ;1 ( ^ ; )c x i = G (x) (3:5) were k = F ;1 ( ;1 ), c =(EY 1 ) 1= and G is te Weibull distribution function, G (x) = 8 < : 1 ; exp f;x g if x, if x<. (3:6) From te observations fy n n = 1 ::: g we tus obtain te estimator from (3.2), te corresponding estimator, ^ and, ^a = ;;1 log ^ (3:7) of te CAR(1) coecient a. Provided te distribution function F of te noise terms Z n in te discrete-time sampled process satises te conditions indicated above, we can also determine te asymptotic distributions of tis estimator. In particular, using a Taylor series approximation, we nd tat i lim P (;)e ;a c k ;1 ^a ; a x = G (x) (3:8)!1 were G is given in (3.6). Since var(y )= 2 =(2a), we use te estimator, to estimate 2. ^ 2 = 2^a X i= (Y i 4. Estimating te Levy increments. ; Y )2 (3:9) So far, we ave made no assumptions about te driving Levy process except for nonnegativity and existence of EL(1) 2. In order to suggest an appropriate parametric model for L and to estimate te parameters, it is important to recover an approximation to L from te observed data. If te CAR(1) process is continuously observed on [ T], ten te argument of Pam-Din-Tuan (1977) can be used to recover fl(t) t T g. His L 2 -based spectral argument wic e applied to Gaussian processes, also applies to te Levy-driven CAR(1) process to give L(t) = ;1 Y (t) ; Y () + a 6 Y (s)ds : (4:1)

7 A direct justication of tis R result can be obtained by dening L as in (4.1) and ten sowing tat Y ()e ;at t + e;a(t;u) dl(u) = Y (t). Tus, using Remark 2 of Section 2, we ave Y ()e ;at + =Y ()e ;at + L(t) ; a =Y ()e ;at + =Y (t)+a =Y (t) as required. e ;a(t;u) dl(u) Y (t) ; Y () + a Y (s)ds ; a e ;a(t;u) L(u)du Y (s)ds Z u ; a e Y ;a(t;u) (u) ; Y () + a e ;a(t;u) Y (u)du ; a 2 Y (s) s e ;a(t;u) du ds (n;1) Y (s)ds du From (4.1), te increment of te driving Levy process on te interval ((n ; 1) n] is given by Z n M L n := L(n) ; L((n ; 1)) = Y ;1 (n) ; Y ((n ; 1))+a Y (u)du : Replacing te CAR(1) parameters by teir estimators and te integral by a trapezoidal approximation, we obtain te estimated increments, M ^L n =^ ;1 Y n ; Y n;1 +^a (Y n i + Y n;1)=2 : (4:2) 5. Te gamma-driven CAR(1) process In tis section, we illustrate te preceding estimating procedure in te case wen L is a standardized gamma process. Tus L(t) as te gamma density f L(t) wit exponent t, scale-parameter ;1=2, mean 1=2 t and variance t. Te Laplace transform of L(t) is ~f L(t) (s) :=E exp(;sl(t)) = exp f;t(s)g <(s) (5:1) were (s) = log(1 + s), = ;1=2 and >. Based on te -spaced observations fy n n= 1 ::: g, we estimate te discretetime autoregression coecient and te CAR(1) parameters a and 2 using (3.4), (3.7) and (3.9) respectively. We ten estimate te Levy increments as in (4.2) and use tem to estimate te parameter of te standardized gamma process L. To obtain te asymptotic distributions of and ^a as! 1 wit xed, we rst sow tat te 7

8 distribution function F of Z n in (3.1) is regularly varying at zero wit exponent and ten determine te coecients k = F ;1 ( ;1 ) and c = (EY 1 ) 1= in (3.5). To do so, we use te Laplace transform (5.1) to investigate te beavior of te density of R R Z 1 = e;a(;t) dl(t) = R e;at dl(t) near zero. Dene W := Z 1 = = e;at dl(t). Te Laplace transform of W is ~f W (s) = exp =exp ; ; Z Z (se ;at )dt log(1 + se ;at )dt Te exponent in (5.2) as te power series expansion, Z Z ; log(1 + se ;at )dt = ; log se ;at (1 + 1 se ) dt ;at log(s) ; a2 + as s!1. Hence ~ f W (s) aste corresponding expansion, : (5.2) 1 sa (1 ; ea ) ; s 2 a (1 ; e2a )+ ~f W (s) ; s e 1 2 a2 + C 1 s + C s were C 1 C 2 ::: are constants depending on,, and a. Since ~ f Z1 (s) = ~ f W (s), s ~ fz1 (s)! () ; e 1 2 a2 as s!1: of Z 1 as te expansion, in a neig- By Teorem 3.2 of Doetsc (1974), te density f Z1 bourood of zero, f Z1 (x) = (); x ;1 e 1 2 a2 + (x) C 1 ; ;( +1) + (x)+1 C 2 ;( +2) + : So and f Z1 (x) x ;1! (); e 1 2 a2 =; as x! F Z1 (x) x () ; e 1 2 a2 =;( +1) as x! : (5:3) Tus te distribution F of Z n is regularly varying at zero wit exponent. From te denition of k in (3.5) we ave 1 = R k F Z1 (du): Tis equation, togeter wit (5.3), gives k ;1 ();1 [;( +1)] ;1= e 1 2 a 1= as!1: (5:4) 8

9 P In order to calculate c, we need to nd E[Y n ], were Y n 1 = j= j Z n;j. Te Laplace transform of Y n is ;sy n ~f Y (s) =Ee = n 1Y j= Ee ;sj Z n;j : So and ence log f ~ Y (s) = n " ~f (s) = exp Y n a =exp = 1X 1X j= j= = ; log ~ f Z1 (s j ) log ~ f W (s j ) 1X j= Z log(1 + s j e ;ay )dy 1X # dilog(1 + s j ) ; dilog(1 + s j e ;a ) j= a dilog(1 + s) R x were dilog is te dilogaritm function, dilog(x) = log(u)=(1 ; u)du. Using Teorem 1 2.1ofBrockwell and Brown (1978), we get E[Y n ] 1 = ;(1 ; ) = a;(1 ; ) Z 1 Z 1 s ; D ~ fy n s ;;1 exp were Df denotes te derivative of f. Ten c = evaluated from (5.5) for xed. (s) ds a dilog(1 + s) log(1 + s)ds (5.5) E[Y n ] i 1= can be numerically Teorem 1. For a sequence of observations fy n n= 1 ::: g from a gamma-driven CAR(1) process, we ave ^a! a a.s. and were G is as in (3.6), =, ^a evaluated troug (5.5). lim P (;)e ;a k ;1!1 (^a ; a)c x is dened in (3.7), k;1 i = G (x) is given in (5.4), and c is Proof. At te beginning of Section 3, weave sown tat Y n is a stationary discrete-time AR(1) wit autoregression coecient 2 ( 1) and i.i.d. noise sequence fz n g. According 9

10 to (5.3), te distribution function F of Z n is regularly varying at zero wit exponent = and satises te condition F () =. Since Z n (L(n) ; L((n ; 1)), R u F (du) < 1 for all >. By Corollary 2.4 of Davis and McCormick (1989), we ave ^! a.s., wic implies ^a! a a.s.. From te same corollary, we also conclude tat lim P!1 k ;1 ( ^ ; )c x i = G (x) were ^ and k;1 are given in (3.4) and (5.4) respectively, andc is evaluated troug (5.5). Using a Taylor series expansion, we nd from tis result tat i lim P (;)e ;a k ;1!1 (^a ; a)c x = G (x): Teorem 1 gives te limiting distribution of for xed as!1.itisofinterest also to consider te beaviour of te estimator as also goes to zero. For any non-negative random variable Y wit density function f(u), we ave Z 1 1=s [EY s ] 1=s = u f(u)du s = 1+s! exp Z 1 u ;1 f(u)du as long as EY ;1 is nite. Applying tis result to Y n and Brown (1978), we obtain lim! ; c = exp E(Y n ) ;1 =exp Z 1 = exp Z 1 1=s u s;1 f(u)du ; = exp EY ;1 as s! e a dilog(1+s) ds and using Teorem 2.1 of Brockwell Z 1 ~f Y (s)ds n : (5.6) Te beaviour of k ;1, dened in (5.4), is more complicated. Using L'Hospital's Rule, we ave log ;(s +1) ; (s +1) lim ; = ; lim s! s s! ;(s +1) = ;; (1) = E were E is te Euler-Masceroni constant, wit numerical value of Hence lim! [;( + 1)] ;1= = e E and k ;1 ();1 e E 1= as!1and! : (5:7) Wen is small, k ;1 and c can be well approximated by (5.6) and (5.7). Since te rate of convergence in Teorem 1, as indicated by k ;1, increases as decreases and since te limiting distribution G becomes degenerate as!, tis suggests te possibility of 1

11 super-convergence of ^a for any xed T >, ^a T= to a as!1and!. In fact, in Section 6, we sow tat! a a.s. as!. Example 1. We now illustrate te estimation procedure wit a simulated example. Te gamma-driven CAR(1) process dened by, DY (t)+:6y (t) =DL(t) t 2 [ 5] (5:8) was simulated at times :1 :2 ::: 5, using an Euler approximation. Te parameter of te standardized gamma process was 2. Te process was ten sampled at intervals = :1 = :1 and = 1 by selecting every 1 t, 1 t and 1 t value respectively. We generated 1 suc realizations of te process and applied te above estimation procedure to generate 1 independent estimates, for eac, of te parameters a and. Te sample means and standard deviations of tese estimators are sown in Table 1, wic illustrates te remarkable accurady of te estimators. Table 1. Estimated parameters based on 1 replicates on [ 5] of te gamma-driven CAR(1) process (5.8) wit =2,observed at times n n = ::: [T=]. Gamma increments Spacing Parameter Sample mean Sample std deviation of estimators of estimators =1 a =.1 a =.1 a To estimate te parameter of te driving standardized gamma process, te following procedure was used. For eac and eac realization, te estimated CAR(1) parameters were used in (4.2) to generate te estimated increments L n n =1 ::: 5=. Tese were ten added in blocks of lengt 1= to obtain 5 independent estimated increments of L in one time unit. Te istogram of te increments for one realization wit = :1 is sown, togeter wit te true probability density of L(1), in Figure 1. Even if we did not know tat te background driving Levy process is a gamma process, te istogram strongly suggests tat tis is te case. For eac and for eac realization of te process, te sample mean ^ of te estimated increments per unit time was ten used to estimate te parameter of te driving standardized Levy process, giving a set of 1 independent estimates of for eac. Te sample means and standard deviations of tese estimators 11

12 are sown in Table L Figure 1: Te probability density of te increments per unit time of te standardized Levy process wit = 2 and te istogram of te estimated increments from a realization of te CAR(1) process (5.8), obtained by computing L (:1) n n =1 ::: 5, from (4.2) and adding successive values in blocks of 1 to give estimated increments per unit time. Table 2. Estimated parameter of te standardized driving Levy process. Spacing Parameter Sample mean Sample std deviation of estimators of estimators = =: =: Estimation for te continuously observed process. It is interesting to note tat from a continuously observed realization on [ T] of a CAR(1) process driven by a non-decreasing Levy process wit drift m =, te value of a can be identied exactly wit probability 1.Tis contrasts strongly wit te case of a Gaussian CAR(1) process. Te result is a corollary of te following teorem. Teorem 2. If te CAR(1) process fy (t) t g dened by (2.1) is driven by a non- 12

13 decreasing Levy process L wit drift m and Levy measure, ten for eac xed t, Y (t + ) ; Y (t) Proof. From (2.6) we nd tat + ay (t)! m a:s: as # : Y (t + ) ; Y (t) =Y ()(e ;a(t+) ; e ;at )+(L(t + ) ; L(t)) ;a e ;a(t;u) (e ;a ; 1)L(u)du ; a + t e ;a(t+;u) L(u)du: Dividing eac side by, letting #, and using te fact tat lim # (L(t+);L(t))= = m (Statland (1965)), we see tat Y (t + ) ; Y (t)! m ; ay ()e ;at + a 2 e ;a(t;u) L(u)du ; al(t) =m ; ay (t): Corollary 1. If m = in Teorem 2 (tis is te case if te point zero belongs to te closure of te support of L(1)), ten for eac xed t, wit probability 1, log Y (t) ; log Y (t + ) a = lim : (6:1) # For eac xed T >, a is also expressible, wit probability 1,as a = sup s<tt log Y (s) ; log Y (t) : (6:2) t ; s Proof. By setting L(t) = for all t in te dening equation (2.1) we obtain te inequality, for all s and t suc tat s<t T, log Y (s) ; log Y (t) <a(t ; s) from wic it follows tat a sup s<tt log Y (s) ; log Y (t) : (6:3) t ; s From Teorem 2witm =we nd tat Y (t) ; Y (t + ) Y (t)! a as # : From te inequalities (6.3) and 1 ; x ;log x for <x 1, we obtain te inequalities, Y (t) ; Y (t + ) Y (t) log Y (t) ; log Y (t + ) a 13

14 and letting # gives (6.1). But tis implies tat wic, wit (6.3), gives (6.2). a sup s<tt log Y (s) ; log Y (t) t ; s Remark 3. If observations are available only at times fn : n = 1 2 ::: [T=]g, and if te driving Levy process as zero drift, Corollary 1 suggests te estimator, ^a T = sup n<[t=] log Y (n) ; log Y ((n +1)) : Tis estimator is precisely te same as te estimator (3.7). Its remarkable accuracy as already been illustrated in Table 1. Te analogous estimator, based on closely but irregularly spaced observations at times t 1 t 2 ::: t suc tat t 1 < t 2 < < t T, is log Y (t n ) ; log Y (t n+1 ) ^a T = sup : n t n+1 ; t n By Corollary 1, bot estimators converge almost surely to a as te maximum spacing between successive observations converges to zero. 7. Conclusions In Section 3 of tis paper, we developed a igly ecient metod, based on observations at times 2 :::, for estimating te parameters of a stationary Ornstein- Ulenbeck process fy (t)g driven by a non-decreasing Levy process fl(t)g. If is small, we used a discrete approximation to te exact integral representation of L(t) in terms of fy (s) s tg to estimate te increments of te driving Levy process, and ence to estimate te parameters of te Levy process. Under specied conditions on te driving Levy process we obtained te asymptotic distribution of te estimator of te CAR(1) coecient as!1wit xed. Te accuracy of te procedure was illustrated wit a simulated example of a gamma-driven process. We also sowed tat te CAR(1) coecient a is determined almost surely by a continuously observed realization of Y on any interval [ T]. Te expression for a suggests an estimator based on discrete observations of Y wic, for uniformly spaced observations, is te same as te estimator developed in Section 3. Te generalization of te procedure to non-negative Levy-driven continuous-time ARMA processes is currently in progress. Acknowledgments Te autors are indebted to te ational Science Foundation for support of tis work under Grant DMS-3819 and PB for te additional support of Deutsce Forscungsgemeinscaft, SFB 386, at Tecnisce Universitat Muncen. 14

15 References Barndor-ielsen, O.E. and Separd,. (21). on-gaussian Ornstein-Ulenbeck based models and some of teir uses in nancial economics (wit discussion). J. Roy. Statist. Soc. Ser. B, 63, 167{241. Brockwell, P. J. and Brown, B.M. (1978). Expansions for te positive stable laws. Z. Warsc. Verw. Gebiete 45, 213{224. Brockwell, P.J. (21). Levy-driven CARMA processes. Ann. Inst. Stat. Mat. 53, 113{124. Brockwell, P.J. and Marquardt, T. (25). Fractionally integrated continuous-time ARMA processes. Statistica Sinica 15, 477{494. Davis, R. and McCormick, W. (1989). Estimation for rst-order autoregressive processes wit positive and bounded innovations, Stocastic Process. Appl. 31, 273{25. Doetsc, G. (1974). Introduction to te teory and application of te Laplace transformation, Springer-Verlag, ew York. Eberlein, E. and Raible, S. (1999). Term structure models driven by general Levy processes, Matematical Finance 9, 31{53. Pam-Din-Tuan (1977). Estimation of parameters of a continuous-time Gaussian stationary process wit rational spectral density function. Biometrika 64, 385{399. Protter, P.E. (24). Stocastic Integration and Dierential Equations. 2nd edition, Springer, ew York. Statland, E.S. (1965). On local properties of processes wit independent increments. Teory of Prob. and Applications 1, 317{