Generalized fractional Ornstein-Uhlenbeck processes

Transcription

1 Generalized fractional Ornstein-Uhlenbeck processes Kotaro ENDO and Muneya MATSUI Department of Mathematics, Keio University Center for Mathematical Science, Munich University of Technology and Department of Mathematics, Keio University arxiv:87.11v1 [math.pr] 14 Jul 8 October 5, 18 Abstract We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU process where the integrand is replaced by the exponential of an independent Lévy process. We call the process the generalized fractional Ornstein-Uhlenbeck (GFOU process. Alternatively, the process can be constructed from a generalized Ornstein-Uhlenbeck (GOU process using an independent fractional Brownian motion (FBM as integrator. We show that the GFOU process is well-defined by checking the existence of the integral included in the process, and investigate its properties. It is proved that the process has a stationary version and exhibits long memory. We also find that the process satisfies a certain stochastic differential equation. Our underlying intention is to introduce long memory into the GOU process which has short memory without losing the possibility of jumps. Note that both FOU and GOU processes have found application in a variety of fields as useful alternatives to the Ornstein-Uhlenbeck (OU process. Keywords: Ornstein-Uhlenbeck processes, Lévy process, Stochastic integral, Long memory, Fractional Brownian motion. 1 Introduction The fractional Brownian motion (FBM is one of the most popular processes for constructing long-range dependent stochastic processes with continuous path and its fields of applications are very wide. To name just a few, we see FBM models in the fields of telecommunications, signal processes, environmental models and economics. A recent reference is e.g., Doukhan et al. (3. Statistical methods for FBM have also been studied (see e.g. Beran (1994. We review the definition and name properties of FBM. Definition 1.1 Let < H 1. A fractional Brownian motion B H := {Bt H} t R is a centered Gaussian process with B H = and Cov ( ( Bt H,Bs H = 1 t H + s H t s H, t,s R. Correspondence to Department of Mathematics, Keio University, Hiyoshi Kohoku-ku, Yokohama 3-85, Japan. mmuneya@gmail.com. This research is supported by JSPS Research Fellowships for Young Scientists. 1

2 From the definition FBM has stationary increments and is self-similar with index H, i.e., for c > {Bct H} d t R = {c H Bt H} t R where = d denotes equality of all finite dimensional distributions. While {Bt H } t R with H = 1/ is a two-sided Brownian motion (BM and has independent increments, {Bt H} t R with H (, 1 (1,1] has dependent increments. For < h < s and t R and N N, Γ h (s := Cov(B t+h B t,b t+s+h B t+s = Cov(B h,b s+h B s ( n 1 h n = (H k s H n (n! n=1 k= ( N n 1 h n = (H k s H n +O ( s H N, as s. (n! n=1 k= Thus, B H with H ( 1,1] has a long memory property, namely, n= Γ h(nh =. Finally B H is knowntohaveboundedp-variationfor1/h < p < (seeproposition.ofmikosch and Norvai sa (. For a more detailed theoretical treatment, we refer to Embrechts and Maejima ( or Samorodnitsky and Taqqu (1994. On the other hand, recently, extensions of the classical OU process have been suggested mainly on demand of applications. The generalized Ornstein-Uhlenbeck process given below is one with Lévy processes and plays important role in Economics (pricing of Asian options, perpetuities and risk theory. For its theories and applications, we refer to e.g, Carmona et al. (1, Erickson and Maller(5, Lindner and Maller(5 and Klüppelberg and Kostadinova(8. A multivariate extension is also considered in Kondo et al. (6 and Endo and Matsui (8. Let {(ξ t,η t } t be a bivariate Lévy process and V be an independent initial random variable. A generalized Ornstein-Uhlenbeck (GOU process is defined as V t = e (V ξt + e ξ s dη s, t. (1 The stationarity and the convergence or divergence property have been intensively studied. If {ξ t } t and {η t } t are independent and e ξ s dη s converges a.s. as t to a finite random variable, V := {V t } t has the stationary version (see e.g. Remark. of Lindner and Maller (5. The short memory property of V was also shown in Section 4 of Lindner and Maller (5. Another extension of the original OU process is the fractional Ornstein-Uhlenbeck process, where FBM is used as integrator. An advantage of using the process is to realize stationary long range dependent processes. Let λ > and an initial random variable X H L 1. A fractional Ornstein-Uhlenbeck (FOU process is defined as X H t = e λt ( X H + e λs dbs H. ( Here we need a non-semimartingale approach to construct stochastic integrals with FBM. We can find several useful theoretical tools in e.g., Lin (1995, Mikosch and Norvai sa ( or Pipiras and Taqqu (. Cheridito et al. (3 has shown that the FOU process is the unique continuous solution of a Langevin equation: X t = X H λ X sds+bt H, t and investigated its dependence properties. The main purpose of this paper is to construct a version of the GOU process which allows for long memory modeling by the use of a FBM.

3 as In order to define a generalized Ornstein-Uhlenbeck process we define a two-sided Lévy process ξ t := { ξ 1 t if t ξ t if t <, (3 where {ξ 1 t} t and {ξ t} t are independent copies of {ξ t } t. We work throughout with a bivariate complete probability space (Ω := Ω 1 Ω, F := F 1 F,P := P 1 P. (4 Let {ξ t } t R defined on (Ω 1, F 1,P 1 be a Lévy process and a FBM {Bt H} t R with Hurst index H (,1 defined on (Ω, F,P which is independent of {ξ t } t R. A generalized fractional Ornstein-Uhlenbeck (GFOU process with initial value Y L 1 (Ω is defined as Y t := e (Y ξt + e ξ s dbs H, t. (5 If the initial variable satisfies Y = then, for convenience, we sometimes replace Y = {Y t } t with Y t := e ξt e ξ s db H s, (6 e ξ s db H s. (7 The process Y is regarded as an extension of V given in (1 where the stochastic process of integration {η t } t is replaced with a {Bt H} t with H (,1 and also is regarded as an extended version of the FOU process where the integrand is replaced by the exponential of an independent Lévy process ξ t. We should remark that Y has jumps caused by the process e ξt. The paper is organized as follows. In Section.1 we recall the definition of Lévy processes and summarize properties needed. In Sections..1 and.. we review Riemann-Stieltjes integrals for functions with bounded p-variation and the stochastic integral in the L (Ω-sense respectively. In Section 3 we investigate the existence of the integral in the GFOU process in order to justify the definition of the GFOU process. The stationarity condition and the second order behavior of the GFOU process are discussed in Section 4, and we observe the long memory property. Here we also examine stochastic integrals constructed by a single FBM, where ξ in the process Y is replaced with B H used as the integrator. In Section 5 we obtain a stochastic differential equation, whose solution is given in form of the GFOU process. We use the following notations throughout. Write a.s. = if equality holds almost surely. We will take the expectations for a bivariate process {(Zt 1,Z t } t R. If the expectations only for a process {Zt} 1 t R is considered, we write its expectation as E Z 1. Preliminaries.1 Lévy processes In this subsection we introduce the setup for the Lévy process. Let ξ := {ξ t } t be a Lévy process on R with (a ξ,ν ξ,γ ξ generating triplet, where a ξ and γ ξ R are constants and a measure ν ξ 3

4 on R\{} satisfies R\{} (1 x ν ξ (dx <. We call ν ξ the Lévy measure of ξ. Then, the characteristic function of ξ t at time t = 1 is written as [ Ee izξ 1 = exp a ] ξ ( z +iγ ξ z + e izx 1 izx1 { x 1} νξ (dx, z R. (8 R\{} For more on Lévy processes and their properties, we refer to Sato (1999. In later sections we consider several examples related the α-stable Lévy motion with index < α <, denoted by ξ α := {ξt α} t. It is a Lévy process and its generating triplet is (,ν α,γ α where { c1 x ν α (dx := 1 α dx on (, c x 1 α dx on (, with c 1, c and c 1 +c. In order to define the in (5, the variation of the Lévy process plays an important role. We give a brief summary based mainly on Section 5.4 of Dudley and Norvaiša (1998 and p.48 of Mikosch and Norvai sa (. Define the p-variation for < p < of a process X := {X t } t R on [t 1,t ] for t 1 < t in R as v p (X := v p (X,[t 1,t ] := sup Xsi X si 1 p, (9 where is a partition t 1 = s < s 1 < < s n = t of [t 1,t ] and n 1. If v p (X,[t 1,t ] < for all t 1 < t, we say X has bounded p-variation, and if v 1 (X,[t 1,t ] < we say it is of bounded variation. Since every Lévy process is a semimartingale v p (ξ < for p (see Lépingle (1976. We will state three useful results which characterize p-variation of Lévy process in terms of the Lévy measure. Unfortunately we can not find a result which uniformly characterize the variation in terms of the Lévy measure. Assumptions and results are somewhat different from paper to paper. The first one is a well-known result (e.g. Theorem 1,9 (i of Sato ( Bounded variation A Lévy process {ξ t } t is of bounded variation if and only if a ξ = and (1 x ν ξ (dx <. R\{} The next result is a combination of Theorems 4.1 and 4. of Blumenthal and Getoor (1961 and Theorem of Monroe (197..(a p-variation of Lévy processes Define β = inf{α > : x <1 x α ν ξ (dx < }. We call β the Blumenthal and Getoor index. If the Lévy process {ξ t } t has no Gaussian component (a ξ =, then i=1 v p (ξ;[,1] < a.s. if p > β v p (ξ;[,1] = a.s. if p < β. (1 4

5 The result by Bretagnolle (197 is a sharpened version of.(a but with zero mean assumption..(b p-variation of Lévy processes Let 1 < p < and {ξ t } t be a Lévy process without Gaussian component (a ξ =. Then v p (ξ;[,1] < a.s. if and only if (1 x p ν ξ (dx <. Otherwise v p (ξ;[,1] = a.s. R\{} In particular for α-stable Lévy processes we have the result by Fristedt and Taylor (1973 which was stated in convenient form in Mikosch and Norvai sa (. 3. p-variation of α-stable Lévy processes Let {ξ α t } t be α-stable Lévy motion. Assume that γ α = for α < 1 and that the Lévy measure is symmetric for α = 1. Then v p (ξ α is finite or infinite with probability 1 according as p > α or p α. For the existence of the infinite interval integral in {Y t } t given in (7 we further need the a.s. behavior of {ξ t } t as t. Our assumption is that lim t ξ t = +. Doney and Maller ([Theorem 4.4] have obtained an equivalent condition of this in terms of the Lévy measure ν ξ. Since several paperswell explain equivalent conditions, (see p.7of Erickson and Maller (5 a.s. or p.174 of Lindner and Maller (5 we do not mention it. Actually if lim t ξ t = + holds, we can assert a stronger result, which is more useful for our aim. Lemma.1 Suppose lim t ξ t a.s. = +. Then for almost all ω 1 Ω 1 there exist δ > and t = t (ω 1 < such that ξ t > δt for t t. The proof is a combination of Theorem 4.3 and 4.4 in Doney and Maller (. Concerning the integral e ξ s dη s in the GFOUprocess given by (1, Erickson and Maller (5 have characterized theconvergence ofimproper integral e ξ s dη s intermsofthelévymeasure of{(ξ t,η t } t, a.s. in which the condition lim t ξ t = + was used.. Integrals with respect to functions with unbounded variation..1 Riemann-Stieltjes integrals with p-variation We review several useful definitions of integrals of functions which have unbounded variation but bounded p-variation. The excellent introduction to this area is given by Dudley and Norvaiša (1998. Let f and g be real functions on [a,b]. Define κ = {u 1,...,u n } to be an intermediate partition of = [a = s < s 1 < < s n = b] given as in (9, namely, s j 1 u i s j for i = 1,...,n. A Riemann-Stieltjes sum is defined as S RS (f,g,,κ := f(u i [g(s i g(s i 1 ]. i=1 5

6 Then we say that the Riemann Stieltjes integral exists and equals to I, if for every ǫ >, there exists δ > such that S RS (f,g,,κ I < ǫ for all partition with mesh max (s i s i 1 < δ and for all intermediate partitions κ of. The following theorem was proved by Young (1936. (See also Theorem.4 of Mikosch and Norvai sa ( or Theorem 4.6 of Dudley and Norvaiša (1998. Theorem.1 Assume f has bounded p-variation and g has bounded q-variation on [a, b] for some p,q > with p 1 +q 1 > 1. Then the integral b fdg exists in the Riemann-Stieltjes sense, and a the inequality b fdg C p,q(v p (f 1/p (v q (g 1/q a holds with C p,q = n 1 n (p 1 +q 1... Integral with respect to FBM in L -sense Another definition of the integral is in L (Ω -sense. Stochastic integrals with respect to FBM is sometimes defined as the L (Ω -limits of integrals of step functions (see e.g. Lin (1995. We see this when B H is the Brownian motion B 1/. If a function f(u : R R satisfies f(u L (R, there exist step functions f n (u := f i 1 {ui <u u i+1 }, < u 1 <... < u n+1 <, f i R, n N. i=1 such that f n converges to f in L (R. Then the integral R f(udb1/ u integrals of step functions R f n (udb 1/ u = i=1 ( f i Bu 1/ i+1 Bu 1/ i, is the L (Ω -limit of the since L (R and L (Ω are isometric and their inner products are equal, namely, [ ( ] E (f n (u f(udbu 1/ = (f n (u f(u du. R R Pipiras and Taqqu ( have investigated a similar characterization for the integral of B H when H 1. We apply this to the existence of the improper integral in the GFOU process {Y t} afterward. Define the linear space { } Sp(B H := X : f n (udbu H X in L (Ω for some sequence (f n n N (step functions. R Pipiras and Taqqu ( have analyzed the functional space of the integrand f(u in which it can be asymptotically approximated by f n (u and R f(udbh u is well-defined. For H (, 1 they succeeded in specified a Hilbert space of functions on the real line which is isometric to Sp(B H. However, for H ( 1,1 they had difficulty in finding the corresponding isometric space, and 6

7 as second best they analyzed inner product spaces in which the integral with B H (H ( 1,1 is well-defined. We give only one such inner product space and its inner product for B H with H ( 1,1. Other inner product spaces do not seem to work for our purpose since they require e.g. characteristic function of f or fractional derivative of f which do not exists in our case where f(u = e ξu. (See Section 7 of Pipiras and Taqqu (. The space is Λ H = { f : f, f Λ H < } ( 1, H,1, (11 where f,g Λ H = c H is the inner product with c H := H(H 1. 3 Existence of the integral R R f(ug(v u v H dudv In this Section we analyze the existence of the integral in the GFOU process given in (5. The definition of the GFOU process includes an integral with respect to FBM. Since paths of FBM are of infinite variation and since FBM with H 1 is not a semimartingale, the stochastic integral with respect to FBM (H 1 is not an Itô integral. Additionally, the integrand of the GFOU process is random and the infinite interval integral (6 is needed for its stationarity. We apply two approaches of the integral in Section. in order to cope with these problems. Proposition 3.1 Let B H := {Bt H} t R be a FBM with H (,1 and ξ := {ξ t } t R be an independent two-sided Lévy process. Assume that {ξ t } t R has bounded p-variation for < p <. Then the integral eξ s dbs H, < t < exists in the path-wise Riemann-Stieltjes sense if 1 +H > 1. p Furthermore with q > 1 and C H p,q = +q 1 n 1 we have n (p 1 ( e ξ s dbs H C p,q sup e ξs (v p (ξ 1/p( ( v q B H 1/q P a.s. s [,t] The proof of Proposition 3.1 is obvious from the continuity of the exponential function and we omit it. Remark 3.1 If p = 1 in Proposition 3.1, ξ has bounded variation and we can define path-wise integrals for all B H with H (,1. On the other hand if H > 1, we can define the path-wise integral for any integrands ξ since v p (ξ < for p >. The reason why we need the path-wise definition besides the L (Ωp-approach is that we want to define the integral for H (, 1, which we could not obtain in the L (Ω-approach. It also gives several useful tools easily, such as integration by parts or chain rule for analyzing stochastic differential equations in Section 5. Example 3.1 Set ξ := ξ α be an α-stable Lévy motion with index α defined in Subsection.1. Then under assumption in 3. p-variation of α-stable Lévy processes and the assumption 1 +H > 1. the integral α e ξα t eξα u db H u exists in the path-wise sense. Hence the GFOU process is well-defined. Both ξ α and B H have infinite variation and the former is known to be extremely heavy tailed process. 7

8 Recall that we write Y t = Y t (ω 1,ω to emphasis that it is a function from probability space (4 and that we write Y t if initial value Y satisfies (6. Define approximating step functions of the integrand e ξt+ξ u of Y t as f t,n (u;ω 1 := f i (u;ω 1 1 {u n i 1 <u u n i }, i=1 where f i (u;ω 1 := e ξt(ω 1+ξ u n (ω 1 i 1. Here u n i s are points in [ N n,t] such that N n = u n < u n 1 < < u n n = t and as n max (u n i u n i 1 and N n. By integrating f t,n with respect to Bt H, we also define approximating sequence as Z n t (ω 1,ω := = f t,n (u;ω 1 db H u (ω f i (u;ω 1 (Bu H n i (ω BH u (ω. n i 1 i=1 In the following theorem we define the integral e ξt ξ u db H u as the limit in probability of the Z n t as n. The reason why we need this approach is that with only path-wise definitions we find difficulty to treat improper integrals. For the existence of the improper integral we should consider long time (t behavior of both {B H t } t R and {ξ t } t R path-wisely, which seems to be not an easy task. Additionally, this approach is well-matched with the analysis of the second order behavior. Theorem 3.1 Let {Bt H} t R be a FBM with H ( 1,1 and {ξ t} t R be an independent two-sided a.s. Lévy process. If lim t ξ t = +, then for each t Zt n given in (1 converges in probability to a limit defined by Y t and which does not depend on the sequence u n i. If further E[e ξ 1 ] < 1, then Zt n converges to Y t in L (Ω and it follows that [ ] E[Y t ] = E e (ξt ξu dbu H = and for < s t, E[Y s Y t ] = s E ξ [e ξs+ξ u ξ t+ξ v ] u v H dudv. (1 Remark 3. (a For H (, 1 we could not validate the existence of the improper integral in Y t. a.s. (b Under the only condition that lim t ξ t = +, we cannot prove the L (Ω convergence. (c The condition E[e ξ 1 a.s. ] < 1 implies lim t ξ t = +, which is shown in Proposition 4.1 of Lindner and Maller (5. Proof of Theorem 3.1. We check that for each ω 1 the integrand e ξt(ω 1+ξ u (ω 1 Λ H given in (11. Since ξ t (ω 1 is constant, we drop it and only show e ξ u (ω 1 Λ H. The function e ξ u (ω 1 is càglàd and bounded 8

9 on any finite interval. Due to Lemma.1 and to the symmetry of two-sided Lévy processes there exists T(ω 1 < such that for all u T(ω 1, ξ u < δu where δ is some positive constant. Then for each ω 1, e ξ u (ω 1 +ξ v (ω 1 u v H dudv < is obvious. Hence we can utilize L (Ω -integral theory in Section... Namely, for each t and for each fixed ω 1, Z n t (ω 1, converges in L (Ω,P. Moreover Z n t converges in probability on (Ω,P for each t since sequence Z n t satisfies the Cauchy criterion, as seen by lim P n,m ( Zn t Zm t > ǫ = lim 1 { Z n n,m t Zt m >ǫ} dp dp 1 Ω 1 Ω = lim P ( Zt n (ω 1 Zt m (ω 1 > ǫdp 1 =. n,m Ω 1 The limit is called Y t and it is F 1 F measurable for each t. Now, with E[e ξ 1 ] < 1, we prove the L (Ω-convergence. We have E[e ξ 1 ] < 1 as well, hence θ 1 := loge[e ξ 1 ] > and θ := loge[e ξ 1 ] >. Then using the covariance of the FBM, we have [ ] E[(Zt n ] = E e ξt+ξ u n +ξ u i 1 n j 1 (B H u n i Bu H n (BH i 1 u n BH j u n j 1 = i=1 i=1 j=1 = c H j=1 [ ] {E ξ e ξt+ξ u n +ξ u i 1 n j 1 1 ( u n i u n j H + u n i 1 un j H + u n i un j 1 H u n i 1 un j 1 H} Γ n (u,v u v H dvdw, (13 where Γ n t (u,v = i=1 j=1 { 1 {u n i 1 <u u n i,un j 1 <v un j,i j} e θ (t u n i 1 θ 1(u n i 1 un j 1 which obviously converges point-wise to +1 {u n i 1 <u u n i,un j 1 <v un j,i<j} e θ (t u n j 1 θ 1(u n j 1 un i 1 }, 1 {u v} e θ (t u θ 1 (u v +1 {u<v} e θ (t v θ 1 (v u. 9

10 Since Γ n t(v,w < M for some M > uniformly in n, t, (13 is bounded by c H M t H. Furthermore according to usual Fubini s theorem, it also follows that E [ (Y t ] ( = Y t (ω 1,ω dp dp 1 Ω 1 Ω [ t ] = c H E ξ e ξt+ξ u +ξ v u v H dudv = c H e θ (t u θ 1 (v u 1 {u v} u v H dudv <. Observe that this integral is finite. Accordingly E[(Zt n ] E [ (Y t ] as n. Now we can apply Theorem of Chung (1 to Zt n and obtain the L (Ω-convergence. In consequence E[Y s Y t ] turns out to be finite and equation (1 follows from Fubini s theorem. Finally E[Zt n ] =, n N implies E[Y t ] =. Hence the proof is complete. The process {Y t } t R obtained in Theorem 3.1 is the GFOU process with initial value Y. We close this section with the following concluding Remarks. Remark 3.3 (a In both Proposition 3.1 and Theorem 3.1 ξ is independent of B H and we have b b e ξt e ξ u dbu H = e ξt+ξ u dbu H, a < b <. a a This is not allowed in usual theory of stochastic integrals related to semimartingale (Protter (4 since {ξ t } t R is not adapted. a.s. (b From Theorem 3.1 lim t ξ t = + implies Y = eξ s dbs H < a.s. Ω. Therefore together with Proposition 3.1 we can also treat {Y t } t = {Y t +Y } t path-wisely. (c We should mention that Erickson and Maller (5[p.81] gave another idea for improper integrals of a function of Lévy processes with respect to FBM: g(ξ t dbt H. Investigation of their idea is also our next concern. 4 Stationarity and Second order behavior of GFOU processes Here we investigate the strict stationarity and the second order behavior of the GFOU process Y := {Y t } t. Since we could not validate the existence of Y with Hurst parameter H (, 1 and since our main concern in this paper is the long memory case, we confine our results to the case H ( 1,1 throughout this section. The stationarity of Y is as follows. Proposition 4.1 If lim t ξ t a.s. = +, then Y t exists for all t and the process Y := {Y t } t is strictly stationary. Proof of Proposition 4.1. Let t 1 < t <... < t m <, m N and h >. We use the sequence Z n t given in (1. Since 1

11 both {ξ t } t R and {B H t } t R have stationary increments, does the pair {(ξ t,b H t } t R as well because of independence. Thus, Z n t i = d = e ξt+ξ u n j 1 (B H u n j Bu H, N n n = u n j 1 1 <... < un n = t i j=1 e ξ t+h+ξ u n +h j 1 (Bu H n j +h BH u n +h j 1 j=1 simultaneously in i and we have = Z n t i 1 +h, 1 i m, (Z n t 1,...,Z n t m d = (Z n t 1 +h,...,zn t m+h. Then by virtue of Theorem 3.1 as n, (Z n t 1,...,Z n t m converges in probability to (Y t1,...,y tn and (Z n t 1 +h,...,zn t m+h converges in probability to (Y t 1 +h,...,y tn+h. This yields (Y t1,...,y tn d = (Y t1 +h,...,y tn+h and the conclusion holds. Remark 4.1 (a The logic in the proof works even in the case H (, 1. If the assumptions of Proposition 3.1 are satisfied and if the integral Y = eξ s dbs H with H (, 1 exists, then Y with H (, 1 is defined and is strictly stationary. (b In connection with the GOU process {V} t given in (1. Proposition 4.1 corresponds to Theorem 3.1 of Carmona et al. (1, where ξ and η are independent, and under conditions of a.s. convergence of the integral e ξ a.s. s dη s and lim t ξ t = + the stationary version exists and equals in distribution to V. Next we investigate the second order behavior of Y and derive the auto-covariance function explicitly. What should be remarked is that while the auto-covariance function of the GOU process V given in (1 decreases exponentially (Theorem 4. of Lindner and Maller (5, that of the FOU process {X H t } t given in (1 decays like a power function (Theorem.4 and Corollary.5 in Cheridito et al. (3. Since Y is regarded as a version of GOU processes and FOU processes, this investigation is interesting. We utilize results in Theorem 3.1 and obtain Theorem 4.1 and Corollary 4.1 below. Note that even the existence of Y and the equation (1 are obtained several difficulties still lay in calculating the auto-covariance function. The integrand in (1 is regarded as exponential moment of of 4 dependent random variable, i.e. E ξ [e (ξs ξ u +ξ t ξ v ] and dependencies of these variables are different in the order of s,u,t and v. We also require that after E ξ [e (ξs ξ u +ξ t ξ v ] is calculated the double integral in (1 has a suitable representation for our purpose. In the proofs of Theorem 4.1 and Corollary 4.1 we will see how to get over these difficulties. Proofs of Theorem 4.1 and Corollary 4.1 are given in Appendix A since they require a lot of technical and tedious calculations. Theorem 4.1 Let {B H t } t R be a FBM with H ( 1,1 and {ξ t} t R be an independent two-sided Lévy process. Suppose that E[e ξ 1 ] < 1. Then the stationary version Y := {Y t } t exists and for 11

12 any s >,t, we have Cov(Y t,y t+s = Cov(Y,Y s ( e θ 1 s = c H Γ(H 1 e θ 1s Γ(H 1 θ θ1 H 1 θ1 H + e θ 1s s H 1 θ 1 (H 1 1 F 1 (H 1,H;θ 1 s+ eθ1s Γ(H 1,θ θ H 1 s [ e θ 1 s = c H Γ(H 1 e θ 1s Γ(H 1 θ θ1 H 1 θ1 H + sh 1 n+1 { (H k θ 1 (H 1 n= k=1 1 (θ 1 s (n+1 ( θ 1 s (n+1γ(n+1,θ 1s n! where θ 1 := log(e[e ξ 1 ] > and θ := log(e[e ξ 1 ] >. Here γ(, and Γ(, are incomplete gamma functions in 8.35 of Gradshteyn and Ryzhik ( and 1 F 1 (, ; is the confluent hyper-geometric function in 9.1 of Gradshteyn and Ryzhik (. Note that since γ(n+1,sθ 1 Γ(n+1 = n! as s, we have Cov(Y t,y t+s = H n=1 n 1 k=1 = O(s H. (H kθ n 1 s H n +O(e θ 1s This conclude that Y with H ( 1,1 is a long memory process. While we obtained Cov(Y t,y t+s using special functions in Theorem 4.1, it is mainly for numerical purpose since for such functions useful softwares are available. Next we investigate long time dependence of {Y t } t with the initial value Y := X L (Ω where X is independent of {ξ t } t and {B H t } t. Corollary 4.1 Let Y := {Y t } t, H ( 1,1 be a GFOU process with the initial value X L (Ω, where X is independent of ξ := {ξ t } t and B H := {B H t } t. Then for fixed t as s. Cov(Y t,y t+s = H n=1 n 1 k=1 = O(s H. }], (H kθ n 1 {s H n e θ 1t (t+s H n }+O(s H N We see what happens to the second order behavior of the process Y if ξ in Y is replaced with B H which is the same process as the variable of integration. Although we expect long memory this does not hold. Note that we need only the probability space (Ω, F,P here. For H ( 1,1 and the initial random variable X L (Ω independent of {Bt H} t R, define W t = e BH t ( X + e BH u db H u. To analyze {W t } t we use the path-wise integral theory (see Subsection..1. Let f be continuous differentiable and F(x = F(+ x f(ydy. Then with H (1,1 it follows that F(Bt H F(BH = f(bt H dbh u 1 a.s.

13 By setting f = e t in above we obtain W t = 1+e BH t (X 1. Proposition 4. Let H ( 1,1 and t,s. Define M 1 := (E[X] 1 and M := E[(X 1 ]. The process {W t } t has the following auto-covariance and correlation functions. } Cov(W t,w t+s = e 1 {th +(t+s H } {M e 1 {th +(t+s H s H} M 1 ( = O e 1 {(t+sh +s H 1 } as s. Corr(W t,w t+s = M e 1 (th s H M 1 e 1 (t+sh M e th M 1 M M 1 e (t+sh We also consider the drift added process = O(e 1 sh as s. Ŵ t = e (BH t +at (X + e BH u +au d(b H u +au, where a > and X L (Ω is independent of {B H t } t. Even if a drift is added, usual path-wise integral works and Ŵ t = 1+(X 1e (BH t +at holds. Then the auto-covariance and correlation functions of {Ŵt} t are calculated is a similar manner and become Cov(Ŵt,Ŵt+s = e a(t+s Cov(W t,w t+s, Corr(Ŵt,Ŵt+s = Corr(W t,w t+s. Thus our conclusion here is that even if a drift is added it does not have long memory. 5 Stochastic differential equation related with GFOU processes We analyze a stochastic differential equation of which a solution is given by the GFOU process. Let U := {U t } t bealévyprocesswithgeneratingtriplet(a U,ν U,γ U. AssumethattheLévymeasure ν U has no mass on (, 1]. The Doléans-Dade exponential of U t is written as E(U t = e ξt where ξ t = U t + a ξ t (log(1+ U s U s. <s t See Section. of Erickson and Maller (5. Here ξ t is the Lévy processes. 13

14 Proposition 5.1 Under the assumption in Proposition 3.1, GFOU {Y t } t with the initial value Y L 1 (Ω satisfies the stochastic differential equation; where E(U t = e ξt. dy t = Y t du t +db H t, (14 Proof of Proposition 5.1. Since the condition of Theorem.1 is satisfied, the integral BH s also exists in the Riemann- deξs Stieltjes path-wise sense. We use the integration by parts formula to Y t and obtain Y t = e ξt (Y B H s deξs +B H t. If we put Q t := Y t B H t, the equation above becomes Q t = e ξt (Q B H s deξs, where Q = Y. Since {e ξt } t is a semimartingale and {Bt H} t is continuous and adapted, the process {Q t } t is also semimartingale. We set R t := e ξt and S t := Q BH s. Then the deξs integration by parts formula for semimartingales (e.g. Corollary, II of Protter (4 yields Q t Q = R t S t R S = = = R s ds s + + e ξ s B s de ξs + Q s du s Observe the relation between e ξt and U t ; + S s dr s +[R,S] t R S + Q s e ξ s de ξs + B H s ( e ξ s de ξs d[e ξ,e ξ ] s. B H s d[e ξ,e ξ ] s Using this we obtain which is equivalent to The proof is now complete. 1 = e ξs e ξs = = + + e ξ s de ξs + du s + Q t Q = Y t Y = e ξ s de ξs +[e ξ,e ξ ] t e ξ s de ξ s +[e ξ,e ξ ] t. + ( Qs +Bs H dus, Y s du s +B H t. 14

15 Remark 5.1 The Lévy measure of {ξ t } t is obtained from that of {U t } t ; ν ξ ((x, = ν U ((,e x 1 and ν ξ ((, x = ν U ((e x 1,. (15 SeeagainSection.ofErickson and Maller (5. Henceifν U isconcretelygiven, using criterion of p-variation in Section.1 we can check the condition of Proposition 3.1. The following technical Lemma is not difficult but useful for the existence of {Y t } t which is directly constructed from the stochastic differential equation (14. The poof is only a calculation and we omit it. Lemma 5.1 Assume that {U t } t is a Lévy process and that {ξ t } t satisfies E(U t = e ξt. Then for < δ <, convergence and divergence of x δ ν ξ (dx and x δ ν U (dx are equivalent. x <1 Example 5.1 As an example we consider the stochastic differential equation (14, where U t is given by an α-stable Lévy motion ξt α (see Section.1. From remark above the Lévy measure ν ξ is given by { c (1 e ν ξ (dx = x 1 α e x dx on (, c 1 (e x 1 1 α e x dx on (,. Observe that x <1 ν ξ (dx x 1 α dx as x and hence variation property of ξ t is the same as that of U t = ξ α t. As a result v p (ξ is finite if p > α. A Proofs of Section 4 Proof of Proposition 4.1. From the stationary version Y is definable. By virtue of the stationarity of Y and Fubini s theorem, we have Cov(Y t,y t+s = Cov(Y t,y t+s = c H E ξ [e (ξs ξ u ξ v ] u v H dudv +c H := I+II. s E ξ [e (ξs ξ u ξ v ] u v H dudv First we consider the integral I. The independent increments property of {ξ t } t R gives E[e (ξs ξ u ξ v ] = E[e (ξs ξu ξv ] = E[1 {u v} e (ξs ξ (ξ ξ u (ξ u ξ v ] +E[1 {u<v} e (ξs ξ (ξ ξ v (ξ v ξ u ] = 1 {u v} E[e (ξs ξ ]E[e (ξ ( u ξ ]E[e (ξ u v ξ ] +1 {u<v} E[e (ξs ξ ]E[e (ξ ( v ξ ]E[e (ξ v u ξ ]. 15

16 The integrand E[e (ξs ξ u ξ v ] is symmetric with respect to u and v, and hence I = c H e θ 1s Then further calculation shows I = c H e θ 1s 1 {u v} e θ u+θ 1 (v u u v H dudv. 1 {u v} e θ u+θ 1 (v u u v H dudv (By change of variables;x = u v = c H e θ 1s e θ u du = c He θ 1s Γ(H 1. θ θ1 H 1 Next we consider II. A similar conclusion as above gives and we have II = c H e θ 1s E[e (ξs ξ v (ξ ξ u ] = E[e (ξs ξv (ξ ξ u ] s e θ 1u (By change of variables;x = v u = c H e θ 1s u e θ 1u s u e θ 1x x H dx (16 = e θ 1(s v+θ 1 (u+v = e θ 1s+θ 1 (u+v (u v H dudv (17 e θ 1(u+x x H dudx (18 = c H e θ 1s 1 { x<u<s x} e θ 1u+θ 1 x x H dx ( s s x = c H e θ 1s e θ1x x H dx e θ1u du+ e θ1x x H dx e θ1u du x s x { s ( = c H e θ 1s 1 1 e θ 1x e θ1x x H dx θ 1 θ 1 ( 1 + e θ1(s x 1 } s θ 1 θ e θ 1x e θ1x x H dx ( = c H e θ 1s 1 s e θ1x x H dx 1 e θ1x x H dx (19 θ 1 θ 1 + eθ 1s e θ1x x H dx θ 1 s ( = c H e θ s 1s H+1 θ 1 (H 1 1 F 1 (H 1,H;θ 1 s 1 Γ(H 1 θ1 H Γ(H 1,θ 1 s + eθ 1s θ1 H 16

17 Combining I and II, we obtain the first assertion. The series representation of the incomplete Gamma function, i.e. gives Γ(α,x = x α 1 e x (1+ Γ(H 1,θ 1 s = We apply the binomial expansion (1 u/s H = to the representation n= n=1 s H 1 θ 1 (H 1 1 x n(α 1 (α n n+1 (H k(θ 1 s (n+1 ( n= k=1 (H (H 3 (H n 1 n! ( u s n, ( < u < s e θ1s s H 1 θ 1 (H 1 1 F 1 (H 1,H;θ 1 s = e θ1s e θ1x x H dx θ 1 (By change of variables;u = s x = sh θ 1 s s e θ 1u (1 u/s H du. Then we exchange the infinite sum and integral by usual Fubuni s theorem and obtain e θ 1s s H 1 θ 1 (H 1 1 F 1 (H 1,H;θ 1 s = sh 1 θ 1 (H 1 n+1 (H k( sθ 1 (n+1γ(n+1,θ 1s. (1 n! n= k=1 Thus substituting these expansions ( and (1 in the previous representation of covariance we obtain the result. Proof of Corollary 4.1. Since X is independent it follows that We divide E[Y t,y t+s ] into piece as follows. E[Y t ]E[Y t+s ] = (E[X] e θ 1(t+s. ( E[Y t Y t+s ] [{ }{ +s }] = E Xe ξt + e (ξt ξu dbu H Xe ξ t+s + e (ξ t+s ξ v dbv H [ +s ] = E[X e (ξt+ξt+s ]+E Xe ξ t+s + e (ξ t+s ξ v dbv H [ ] [ +s +E Xe ξ t+s e (ξt ξu dbu H +E e (ξt ξ u +ξ t+s ξ v dbu H dbh v ]. 17

18 The first term is calculated as E[X e (ξt+ξ t+s ] = E[X ]E[e (ξ t+s ξ t+ξ t ] = E[X ]E[e ξs ]E[e ξt ] = E[X ]e θ 1s θ t. (3 From Theorem 4.1, the second and the fourth terms are. We only need the last term, which is calculated as [ +s ] Cov(Y t,y t+s E e (ξt ξu dbu H e (ξ t+s ξ v dbv H t [ +s ] E e (ξt ξu dbu H e (ξ t+s ξ v dbv H [ ] E e (ξt ξu dbu H e (ξ t+s ξ v dbv H = Cov(Y t,y t+s { e θ1t Cov(Y,Y t+s e θ1(t+s Cov(Y,Y t } e θ 1s E [ e ξt Y Y t ] e θ 1 s E[e ξt Y Y t ], (4 where Y t is Y t with initial value. By adding up (, (3 and (4, we have Cov(Y t,y t+s = Cov(Y t,y t+s e θ1t Cov(Y,Y t+s ( +e θ 1s E[X ]e θ1t (E[X] e θ1t +e θ1t Cov(Y,Y t E[e ξt Y Y t ] E[e ξt Y Yt ]. Hence via Theorem 4.1 the conclusion follows. Acknowledgment: The authors are grateful to Prof. Makoto Maejima for very helpful comments. We give deep thanks to Prof. Claudia Klüppelberg for careful reading our paper and for useful comments. Prof. Alexander Lindner suggested the topic of Section 5. We also thankful to Prof. Jean Jacod for pointing out several theoretical mistakes and for giving us nice suggestions. We shall never forget their kindness. References J. Beran. Statistics for Long-Memory Processes (Monographs on Statistics and Applied Probability. Chapman & Hall, USA, R.M. Blumenthal and R.K. Getoor. Sample functions of stochastic processes with stationary independent increments. J. Math. and Mech., 1: , J. Bretagnolle. p-variation de fonctons aléatoires. Séminaire de Probabilités VI;Lect. Notes in Math., 58:51 71,

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