Analysis of the Rosenblatt process

Transcription

1 Analysis of the Rosenblatt process Ciprian A. Tudor SAMOS/MATISSE, Centre d Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris 1, 9, rue de Tolbiac, Paris Cedex 13, France. June 3, 6 Abstract We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Major (1979, Taqqu (1979. This process is non-gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus. AMS Classification Numbers: 6G1, 6G15, 6H5, 6H7 Key words: Non Central Limit Theorem, Rosenblatt process, fractional Brownian motion, stochastic calculus via regularization, Malliavin calculus, Skorohod integral. 1 Introduction A selfsimilar object is exactly of approximately similar to a part of itself. Selfsimilar processes are invariant in distribution under suitable scaling. They are of considerable interest in practice since aspects of the selfsimilarity appear in different phenomena like telecommunications, economics, hydrology or turbulence. We refer to the work of Taqqu [43] for a guide on the appearance of the selfsimilarity in many applications and to the monographs by Samorodnitsky and Taqqu [39] and by Embrechts and Maejima [13] for complete expositions on selfsimilar processes. In this work we analyze a special class of selfsimilar processes that are limits in the so called Non Central Limit Theorem (see Dobrushin and Major [11] or Taqqu [4]. Let us briefly recall the general context. Consider (ξ n n Z a stationary Gaussian sequence with mean zero and variance 1 such that its correlation function satisfies r(n := E (ξ ξ n = n H k L(n (1 1

2 with H ( 1, 1 and L is a slowly varying function at infinity (see e.g. [13]. Denote by H m(x the Hermite polynomial of degree m given by H m (x = ( 1 m e x d m dx e x m. Let g be a function such that E(g(ξ = and E(g(ξ <. Suppose that g has Hermite rank equal to k; that is, if g admits the following expansion in Hermite polynomials g(x = j c j H j (x, c j = 1 j! E (g(ξ H j (ξ then k = min{j; c j }. Since E [g(ξ ] =, we have k 1. Then the Non Central Limit Theorem ([11], [4] says that 1 n H [nt] g(ξ j j=1 converges as n in the sense of finite dimensional distributions to the process k ZH(t k = c(h, k (s y i ( H k + dsdb(y 1... db(y k, ( R k j=1 where x + = max(x, and the above integral is a multiple Wiener-Itô stochastic integral with respect to a Brownian motion B(y y R (see [7] for the definition. The constant c(h, k is positive and it will be taken such that E ( ZH k (1 = 1. The process (ZH k (t t is called the Hermite process and it is H-selfsimilar in the sense that for any c >, (ZH k (ct =(d (c H ZH k (t, where = (d means equivalence of all finite dimensional distributions, and it has stationary increments. When k = 1 the process given by ( is nothing else that the fractional Brownian motion (fbm with Hurst parameter H ( 1, 1. For k the process is not Gaussian. If k = then the process ( is known as the Rosenblatt process (it has actually called in this way by M. Taqqu in [41]. The fractional Brownian motion is of course the most studied process in the class of Hermite processes due to its significant importance in modeling. A stochastic calculus with respect to it has been intensively developed in the last decade. We refer, among others, to [4], [5], [9], [16]. Our main interest consists here in the study, from the stochastic calculus point of view, of the Rosenblatt process. Although it received a less important attention than the fractional Brownian motion, this process is still of interest in practical applications because of its selfsimilarity, stationarity of increments and long-range dependence. There exists a consistent literature that focuses on different theoretical aspects of the Rosenblatt processes. Let us recall some of these works. For example, extremal properties of the Rosenblatt distribution have been studied by J.M. Albin in [] and [3]. The rate of convergence to the Rosenblatt

3 process in the Non Central Limit Theorem has been given by Leonenko and Ahn [3]. Pipiras [9] and Pipiras and Abry [3] studied the wavelet-type expansion of the Rosenblatt process. A law of iterated logarithm has been given is [15]. Among the applications of the Rosenblatt process in statistics or econometrics, we mention the following. In the unit root testing problem with errors being nonlinear transforms of linear processes with long-range dependence, the asymptotic distributions in the model are shown in [46] to be functionals of Hermite processes. limiting distributions of the parabolically rescaled solutions of the heat equation with singular non-gaussian data have similar behavior to the Rosenblatt distribution (see [4] the Rosenblatt distribution also appears to be the asymptotic distribution of an estimator related to the semiparametric bootstrap approach to hypothesis tests (see [18] or to the estimation of the long-range dependence parameter ([1] Besides these more or less practical applications of the Rosenblatt process, denoted in the following by Z, our motivation is also theoretical; it comes from the recent intensive interest to push further the stochastic calculus with respect to more and more general integrator processes. We believe that this process constitutes an interesting and instructive example where the recent developed techniques of the generalized stochastic calculus can find a significant test bench. We actually use the two principal methods to develop a stochastic integration theory: the pathwise type calculus and the Malliavin calculus/skorohod integration. The first approach (that includes essentially the rough paths analysis, see [34], and the stochastic calculus via regularization, see [36] can be directly applied to the Rosenblatt process because of its regular paths and of the nice covariance structure; a pathwise Itô formula can be written and Stratonovich stochastic equation with Z as noise can be considered. The Malliavin calculus and the Skorohod integration are in general connected in a deeper way to the Gaussian structure of the integrator process and as it will be seen in the present work, Skorohod Itô formula can be derived only in particular cases. Although the formula we obtain is rather complicated and not easily tractable, the principal signification of the result is the fact that one can precisely see here that the Gaussian nature of the integrator process is decisive in the stochastic integration theory; once we go out from the Gaussian context, one cannot obtain Itô s formulas that end by a second derivative term. We organized our paper as follows. Section presents basic properties of the Rosenblatt process. In particular we prove a stochastic integral representation on a finite integral that will be useful for the construction of the stochastic calculus. In Section 3 we introduce Wiener integrals with respect to Z by following the ideas in [5] and []. We define in Section 4 the Hilbert-valued Rosenblatt process and we consider stochastic evolution equations with this process as noise. Section 5 describes the application of the stochastic calculus via regularization introduced by F. Russo and P. Vallois in [36] to the Rosenblatt process and in Section 6 we 3

4 discuss the Skorohod (divergence integral: we define the integral and we give conditions that ensure the integrability and the continuity of the indefinite integral process. In Section 7 we prove the relation between the pathwise and the divergence integrals: here the pathwise integral is equal to the Skorohod integral plus two trace terms (in the fbm case there is only a trace term. Finally Section 8 contains a discussion on the Itô formula in the Skorohod sense. On the Rosenblatt process In this section we will analysis some basic properties of the Rosenblatt process; in particular we are interested in its representation as a stochastic integral on a finite interval. As we said, this process is obtained by taking k = in the relation (, so ( Z (t := Z(t = a(h (s y 1 H + (s y H + ds db(y 1 db(y (3 R R where (B(y, y R is a standard Brownian motion on R. The constant a(h is a positive normalizing constant and it is chosen such that E(Z(1 = 1. It follows actually from [5] that ( β( a(h H =, H 1 1 H(H 1 Recall that the process (Z(t t [,T ] is selfsimilar of order H and it has stationary increments; it admits a Hölder continuous version of order δ < H. Since H ( 1, 1, it follows that the process Z exhibits long-range dependence. Since our main interest consists in the construction of the stochastic calculus with respect to the process Z, the representation (3 is not very convenient; as in the fbm case, we would like to represent Z t as a stochastic integral with respect to a Brownian motion with time interval [, T ]. Recall that the fbm with H > 1 can be written as B H t = with (W t, t [, T ] a standard Wiener process and K H (t, sdw s (4 where t > s and K H (t, s = c H s 1 H (u s H 3 u H 1 du (5 c H = s ( 1 H(H 1 β( H, H 1. (6 Note that to prove the representation (4 (at least in law it suffices to see that the right member has the same covariance R as the fbm; otherwise, it can be easily seen from the expression of 4

5 the kernel K that the right member in (4 is H-selfsimilar with stationary increments and as a consequence it cannot be nothing else but a fractional Brownian motion with parameter H. Since the Rosenblatt process is not Gaussian, the proof in its case of a similar representation to (4 needs a supplementary argument; in fact we have the following Proposition 1 Let K be the kernels (5 and let (Z(t t [,T ] be a Rosenblatt process with parameter H. Then it holds that [ t ] t Z(t = (d d(h y 1 y (u, y 1 KH (u, y du db(y 1 db(y (7 where (B t, t [, T ] is a Brownian motion, and H = H + 1 d(h = 1 ( H + 1 H (H 1 (8 1. (9 Remark 1 i The constant d(h is a normalizing constant, it has been chosen such that E(Z(tZ(s = 1 ( t H + s H t s H. Indeed, E(Z(tZ(s = d(h s ( = d(h s dy 1 dy y 1 y y 1 y dvdu (u, y 1 KH ( u v = d(h (H (H 1 (u, y KH (u, y 1 KH (v, y 1dy 1 u v H dvdu = R(t, s. (v, y 1 KH (v, y dudv ii It can be seen without without difficulty that the process [ t ] t Z (t := d(h y 1 y (u, y 1 KH (u, y du db(y 1 db(y 5

6 defines a H selsimilar process with stationary increments. Indeed, for any c >, ct [ ct ] ct Z (ct = y 1 y (u, y 1 KH (u, y du db(y 1 db(y ct [ ct ] t = y 1c y (cu, y 1 KH (cu, y cdu db(y 1 db(y c = [ y 1 y (cu, cy 1 KH (cu, cy cdu ] db(cy 1 db(cy and since B(cy = (d c 1 B(y and KH (cu, cyi = ch 3 KH (u, y i we obtain Z(ct = (d c H Z(t. The fact that Z has stationary increments follows from the relation for any s, t [, T ], s < t and h >. K H (t + h, s K H (t, s = K H (t s, h Proof of Proposition 1: Let us denote by Z (t the right hand side of (7. Consider b 1,..., b n R and t 1,..., t n [, T ]. We need to show that the random variables n n b l Z(t l, b l Z (t l have the same distribution. l=1 We will use the following criterium by Fox and Taqqu (see [14]: If f L ([, T ] is a symmetric function, then the law of the multiple Wiener-Itô integral I (f is uniquely determined by its cumulants, where the mth cumulant of f is given by c m (f = l=1 (m 1! m f(x 1, x f(x, x 3... f(x m 1, x m f(x m, x 1 dx 1... dx m. (1 R m In other words, if two symmetric functions f, g L ([, T ] have the same cumulants, then the multiple Wiener-Itô integrals of order two I (f and I (g have the same law. We will show that, for every t, s [, T ], the random variables Z t + Z s and Z t + Z s have the same law; the general case case will follow by a similar calculation. It holds that Z t + Z s = I (f t,s where f t,s (y 1, y = 1 [,t] (y 1 1 [,t] (y +1 [,s] (y 1 1 [,s] (y y 1 y y 1 y (u, y 1 KH (u, y du (u, y 1 KH (u, y duv (11 6

7 We have denoting by a m := (m 1! m d(h m, c m (f s,t = a(m f t,s (y 1, y... f t,s (y m, y 1 dy 1... dy m R m = a(m dy 1... dy m R ( m y 1 y (u 1, y 1 KH (u 1, y du 1 + ( y y 3 (u, y KH (u, y 3 du +. (.. y m y 1 (u m, y m KH (u m, y 1 du m + and by classical Fubini theorem c m (f s,t = a(m... t j {t,s} ( u1 u m ( u1 u um 1 u m = a(m t j {t,s} 1 y 1 y y y 3 y m y 1 m... du 1... du m (u 1, y 1 KH (u m, y 1 dy 1 1 m (u 1, y KH (u, y dy 1 (u 1, y 1 KH (u 1, y du 1 (u, y KH (u, y 3 du (u m, y 1 KH (u m, y m du m (u m, y m KH (u m, y m dy m m 1 m 1... m du 1... du m u 1 u H u u 3 H... u m u 1 H. (1 with a(m = a(m (H (H 1 m. The computation of the cumulant of Z t + Z s is similar. s, t [, T ], Z(t + Z(s = I (g s,t where Indeed, we can write, for ( g s,t = a(h (u y 1 H + (u y H s + du + (u y 1 H + (u y H + du 7

8 and the mth cumulant of the kernel g s,t is given by c m (g s,t = b(m dy 1... dy m R ( m t (u 1 y 1 H + (u 1 y H + du 1 + ( (u y H + (u y 3 H + du +... ( (u m y m H + (u m y 1 H + du 1 + = b(m = Since for any a > we get t j {t,s} 1... m (u 1 y 1 H + (u m y 1 H R (u m 1 y m H R R (u y a 1 + du 1... du m + dy 1 R (u 1 y 1 H (u y H + (u m y m H + dy m. c m (g s,t = b(mβ( H, H 1m + (u 1 y H + du 1 + (u y 3 H (u m y m H + du + (u m y 1 H (u 1 y H + (u y H (v ya 1 + dy = β(a, a 1 u v a 1 t j {t,s} 1 + dy m... du 1... du m u 1 u H u u 3 H... u m u 1 H and it remains to observe that a (m = b(m which implies that (1 equals (13. + du m (13 of (7. From now on we will use the version of the Rosenblatt process given by the right side We will finish this section by proving that the Rosenblatt process possesses a similar property to the fbm, that is, it can be approximated by a sequence of semimartingales (here actually, since H > 1, by a sequence of bounded variation processes. In the fbm case, the property is inherited by the divergence integral (see [4], [7], [6]; this fact can be used to construct financial models with the Rosenblatt process as noise (see [6]. The basic observation is that, if one interchanges formally the stochastic and Lebesque integrals in (7, one gets ( t u u Z(t = (u, y 1 KH (u, y db(y 1 db(y du 8

9 but the above expression cannot hold because the kernel KH (u, y1 KH (u, y does not be- (u, y 1 behaves on the diagonal as (u long to L ([, T ] since the partial derivative KH y 1 H. Let us define, for every ε >, [ t Z ε (t = d(h = := ( u u A ε (udu. y 1 y (u + ε, y 1 KH (u + ε, y du (u + ε, y 1 KH (u + ε, y db(y 1 db(y ] db(y 1 db(y Since A ε L ([, T ] Ω for every ε > and it is adapted, it follows that the process Z ε is a semimartingale. Proposition For every t [, T ], Z ε (t Z(t in L (Ω. du Proof: We have Z ε (t Z(t = ( ( t y 1 y db(y 1 db(y (u + ε, y 1 KH (u + ε, y KH (u, y 1 KH (u, y du and E Z ε (t Z(t = ( ( dy 1 dy y 1 y (u + ε, y 1 KH (v + ε, y 1 KH y 1 y dvdu (u + ε, y KH (u, y 1 KH (u, y (v + ε, y KH (v, y 1 KH (v, y ( Clearly the quantity K H (u + ε, y 1 KH (u + ε, y KH (u, y 1 KH (u, y converges to zero as ε for every u, y 1, y and the conclusion follows by the dominated convergence theorem. 9

10 3 Wiener integrals The covariance structure of the Rosenblatt process allows to construct Wiener integrals with respect to it. We refer to Maejima and Tudor [5] for the definition of Wiener integrals with respect to general Hermite processes and to Kruk and al. [] for a more general context. Let us recall the main points and translate this construction in our context. One note that Z(t = I ( 1 [,t] (y1, y db(y 1 db(y where the operator I is defined on the set of functions f : [, T ] R and takes values in the set of functions g : [, T ] R and it is given by I(f(y 1, y = d(h f(u KH y 1 y If f is an element of the set E of step functions on [, T ] of the form (u, y 1 KH (u, y du. (14 n 1 f = a i 1 (ti,t i+1 ], t i [, T ] (15 then we naturally define its Wiener integral with respect to Z as i= n 1 ( T f(udz(u := a i Zti+1 Z ti = i= Let H be the set of functions f such that I(f(y 1, y db(y 1 db(y. (16 It can be seen that f H := I(f(y 1, y dy 1 dy <. (17 f H = d(h = d(h ( f(u KH y 1 y dy 1 dy y 1 y f(uf(v KH (u, y 1 KH ( T u v = d(h = H(H 1 (u, y 1 KH (u, y du dy 1 dy y 1 y dvdu (u, y KH (v, y 1 KH (v, y dvdu (u, y 1 KH (v, y 1dy 1 f(uf(v u v H dvdu. 1

11 It can be proved as in [5] or [] that the mapping f f(udz(u defines an isometry from E to L (Ω and it can be extended by continuity to an isometry from H to L (Ω because E is dense in H (see [3]. We will call this extension the Wiener integral of f H with respect to Z. Remark It follows from Pipiras and Taqqu (see [3] that the space H contains not only functions but its elements could be also distributions. Therefore it is suitable to know subspaces of H that are spaces of functions. A such subspace is H where It actually holds H = {f : [, T ] R f(u f(v u v H dvdu < }. L 1 H ([, T ] H H. The space H (and hence H is not complete with respect to the norm H but it is a Banach space with respect to the norm f H = H(H 1 f(u f(v u v H dvdu. The Wiener integrals f(udz(u and g(udz(u are not necessarily independent when the functions f and g are orthogonal in H. A characterization of their independence is given in the next result. Proposition 3 Let f, g H. Then f(udz(u and g(udz(u are independent if and only if f( KH (, y 1, g( KH (, y H = a.e.(y 1, y [, T ] (18 where H is the space analogous to H corresponding to the Hurst parameter H. Proof: We use a result by Üstunel-Zakai [4] (see also Kallenberg []: two multiple Wiener- Itô integrals with respect to the standard Wiener process I n (f and I m (g with f, g symmetric, f L [, T ] n and g L [, T ] m are independent if and only if f 1 g = a.e. on [, T ] m+n, where (f 1 g(t 1,..., t n 1, s 1,..., s m 1 = f(t 1,..., t n 1, tg(s 1,..., s n 1, tdt. 11

12 and Then Let us apply the above result to (F 1 G(y 1, y = F (y 1, y = 1 [,t] (y 1, y G(y 1, y = 1 [,t] (y 1, y ds = c(h f(u KH y 1 s and the conclusion follows easily. f(u KH y 1 y g(u KH y 1 y (u, y 1 KH f(ug(v KH As an immediate consequence, we obtain (u, y 1 KH (u, y du (u, y 1 KH (u, y du. (u, sdu y s g(v KH (u, y 1 KH (v, y u v H dvdu (v, y KH (v, sdv Corollary 1 If f g = a.e. on [, T ] then the random variables f(udz(u and g(udz(u are independent. Remark 3 The construction of Wiener integrals with respect to the Rosenblatt process allows to consider associated Ornstein-Uhlenbeck processes. This has been done in [5] for general Hermite processes of order k following the argument in [8] for the fbm case. It can be showed that the equation X t = ξ λ X s ds + σz(t, t, (19 where σ, λ > and the initial condition ξ is a random variable in L (Ω has an unique solution that can be represented as X ξ (t = e (ξ λt + σ e λu dz(u, t. where the stochastic integral above exists in the Wiener sense. When the initial condition is ξ = σ eλu dz(u (in [8] the integrals are considered on the whole real line, the solution of (19 can be written as X(t = σ e λ(t u dz(u ( and it is called the stationary Rosenblatt Ornstein-Uhlenbeck process. 1

13 Remark 4 The Non Central Limit Theorem given by [11], [4] can be extended to Wiener integrals (see [5]. More precisely, under suitable assumptions on the deterministic function f H one obtains that the sequence 1 ( j n H f g(ξ j n j Z converges weakly when n, to the Wiener integral f(udz(u (g and ξ j were introduced in Section 1. 4 Infinite dimensional process and stochastic evolution equations In this part we define a Hilbert-valued Rosenblatt process and we consider stochastic evolution equations driven by it. Let us consider U a real and separable Hilbert space and Q a nuclear, self-adjoint positive and nuclear operator on U. There exists then a sequence < λ n of eigenvalues of Q such that n 1 λ n <. Moreover the corresponding eigenvectors form an orthonormal basis in U. We define the infinite dimensional Rosenblatt process on U as Z(t = ν λn e n z j (t (1 where (z j j is a family of real independent Rosenblatt processes. Note that the series (1 is convergent in L (Ω for every t [, T ] since E Z(t = j 1 λ j E(z j = t H j 1 λ j <. Note also that Z has covariance function R(t, s in the sense that for every u, v U, and for every s, t [, T ] E Z(t, u U Z(s, v U = R(t, s Qu, v U. This can be proved exactly as fbm case (see [45]. In some situations the assumption that Q is nuclear is not convenient. For example one cannot take Q to be the identity operator, that is λ n = 1 for every n. Therefore, if n λ n = we will consider a bigger real and separable Hilbert space U 1 U such that the inclusion U U 1 is nuclear. Then the quantity Z(t = j z j (te j ( is well-defined stochastic process in U 1. 13

14 In the sequel we will consider the infinite dimensional Rosenblatt process to be defined by (. Following the one dimensional case, one can introduce Wiener integrals with respect to the Hilbert-valued process Z. Let V be another Hilbert space Let (Φ s, s [, T ] a stochastic process with valued in the space of linear operators L(U, V. We put for every t [, T ] Φ s dz(s = j 1 Φ s e j dz j (s where Φ se j dz j (s is a V valued random variable. Note that the integral exists in L (Ω, V if E Φ s dz(s = Φe j H V <. V j Remark 5 If the integrand Φ does not depend on time, then we find E ΦdZ(s = t Φe n V E dz j (s = t H V j j Φe n V and it can be seen that the integral ΦdZ(s exists if and only if Φ is a Hilbert-Schmidt operator. Now we introduce stochastic evolution equations driven by the infinite-dimensional Rosenblatt process. Let A : Dom(A V V be the infinitesimal generator of the strongly continuous semigroup (e ta t [,T ]. We study the equation dx(t = AX(tdt + ΦdZ(t (3 where X( = x V and Φ L(U; V. We will consider mild solution of (3, that is, (when it exists, it can be written as X(t = e ta x + e (t sa ΦdZ(s. (4 We will not assume that Φ is Hilbert-Schmidt (although the integral ΦdZ exists if and only if Φ is Hilbert-Schmidt; this assumption is unnecessary because, under suitable hypothesis on A, the integral e(t sa ΦdZ(s will exist even when Φ is not Hilbert-Schmidt operator. The method used in the fbm case will allow to prove the next theorem. Theorem 1 Let Z be given by ( with H ( 1, 1. Consider Φ L(U; V and A : Dom(A V V be a negative self-adjoint operator. Then there exists a mild solution X of the equation (3 if and only if the operator Φ G H ( AΦ is a trace class operator, where G H (λ = (max (λ, 1 H. (5 14

15 Remark 6 in [44] in the fbm case is assumed that the spectrum of A, σ(a (, l] with l >. The situation when is an accumulation point of the spectrum is not treated; this case is solved in [1]. Proof: Since E e (t sa ΦdZ(s V = c(h n e (t ua Φe n, e (t va Φe n V u v H dudv and what it follows is a deterministic problem that can be solved as in [44]. If S 1 denotes the unit circle and A is the Laplacian on the circle, we have Corollary Assume that U = V = L (S 1 and A = is the Laplacian on U. Denote by (e n, f n n 1 the eigenvectors of that form an orthonormal basis in L (S 1. Let (q n n be a bounded sequence of non-negative real numbers and Z(t = qn e n z n (t + qn f n z n (t n n with (z j, z j j independent real Rosenblatt processes. Then (3 has an unique mild solution such that X(t L (Ω, V if an only if q n n 4H <. n 5 Pathwise stochastic calculus At this point, we will start to develop a stochastic integration theory with respect to the Rosenblatt process. In general, for processes that are not semimartingales, the Itô s theory cannot be applied. One needs generalized alternative ways to integrate stochastically with respect to such processes. In general these generalized method are essentially of two types: the first is the pathwise type calculus and (here we included the rough path analysis [34] and the stochastic calculus via regularization [36] and the second type is Malliavin calculus and the Skorohod integration theory [7]. In general the pathwise type calculus is connected to the trajectorial regularity and/or the covariance structure of the integrator process. The Malliavin calculus instead is very related to the Gaussian character of the driven process. Since the Rosenblatt process with H > 1 has zero quadratic variation (see [37] and regular paths (Hölder continuous of order H ε, the pathwise calculus can be naturally applied to construct stochastic integrals with respect to it. Here we choose to use the approach of Russo and Vallois. Let us list first the main ingredients of the stochastic calculus via regularization. Let (X t t and and (Y t t continuous processes. We introduce, for every t, I (ε, Y, dx = X s+ε X s Y s ds, I + (ε, Y, dx = ε 15 X s X (s ε+ Y s ds, ε

16 and C ε (X, Y (t = I (ε, Y, dx = X s+ε X (s ε+ Y s ds ε (X s+ε X (s ε+ (Y s+ε Y (s ε+ ds. ε Then the forward, backward and symmetric integrals of Y with respect to X will be given and Y d X = lim ε + I (ε, Y, dx, Y d + X = lim ε + I+ (ε, Y, dx, Y d X = lim ε + I (ε, Y, dx (6 provided that the above limits exist uniformly in probability (ucp. The covariation of X and Y is defined as [X, Y ] t = ucp lim ε + C ε(x, Y (t. If X = Y we denote [X, X] = [X] and when [X] exists then X is said to be a finite quadratic variation process. When [X] =, then X is called a zero quadratic variation process. The Rosenblatt process is clearly a zero quadratic variation process since EC ε (Z, Z(t = E 1 ε (X s+ε X s ds = tε H 1 ε. Therefore the stochastic calculus via regularization can be directly applied to it. Precisely, it follows from Proposition 4. of the Russo and Vallois survey [38] that every f C (R, the integrals f (Xd X, f (Xd + X, exist and are equal and we have the Itô s formula f(x t = f(x + f (Xd X f (Xd X. (7 Remark 7 An immediate consequence of the existence of the quadratic variation of the Rosenblatt process is the existence and uniqueness of the solution of a Stratonovich stochastic differential equation driven by Z. Concretely, if σ : R R and b : [, T ] R R satisfy some regularity assumptions and V is a locally bounded variation process, then the equation dx(t = σ(x(td Z(t + b(t, X(tdV (t (8 with X( = G where G is an arbitrary random variable, has an unique solution (see [37] for the definition of the solution. 16

17 6 Skorohod integral with respect to the Rosenblatt process In this part we define a divergence integral with respect to (Z(t t [,T ]. Constructing generalized Skorohod integrals with respect to processes that are not necessarily Gaussian or semimartingales constitutes a frequent topic. For results in this direction, we refer among others, to [33], [19], [6], [17] or []. We will need some basic elements of the Malliavin calculus with respect to a Wiener process (W t t [,T ]. By S we denote the class of smooth random variables of the form where f C b F = f (W t1,..., W tn, t 1,..., t n [, T ] (9 (Rn. If F is of the form (9, its Malliavin derivative is defined as D t F = n i=1 f x i (W t1,..., W tn 1 [,ti ](t, t [, T ]. The operator D is an unbounded closable operator and it can be extended to the closure of S (denoted D k,p, k 1 integer, p with respect to the norm F p k,p = E F p + k j=1 E D (j F p, F S, k 1, p L ([,T ] j where the jth derivative D (j is defined by iteration. The Skorohod integral δ is the adjoint of D. Its domain is Dom(δ = {u L T ([, T ] Ω/ E u s D s F ds C F } and D and δ satisfy the duality relationship E (F δ(u = E D s F u s ds, F S, u Dom(δ. (3 We define L k,p = L p ( [, T ] ; D k,p. Note that L k,p Dom(δ. We denote δ(u = u sδw s. We will need the integration by parts formula if F D 1, and u L 1,. F δ(u = δ(f u + D s F u s (31 We also mention that the Skorohod integral with respect to the fbm B H with Hurst parameter H > 1 is defined through a transfer operator g s db H s = s 17 g r K H r (r, sdrdw s (3

18 where the integral in the right side above is a Skorohod integral with respect to W. moreover g is Skorohod integrable with respect to b H if the quantity s g r KH r (r, sdr is Skorohod integrable with respect to W. Definition 1 Let us consider a square integrable stochastic process (g s s [,T ]. Following (16 and (3 we define its Skorohod integral with respect to Z by := = g s dz(s I(g(y 1, y db(y 1 db(y ( g(u KH y 1 y (u, y 1 KH (u, y du db(y 1 db(y. (33 We will say that a process g is Skorohod integrable with respect to Z if the process Ig Domδ (, where δ ( is the double Skorohod integral with respect to the Brownian motion B. We refer to [8] for the study of double (and multiple Skorohod integrals. Remark 8 Note that the Skorohod integral coincide with the Wiener integral if the integrand g is a deterministic function in H. Another Skorohod integral with respect to Z has been introduced in [] as the adjoint of some Malliavin derivative with respect to Z but this integral does not coincide with the Wiener integral for deterministic integrands. Next lemma gives a condition that ensures the Skorohod integrability. Lemma 1 Let g L (Ω; H be such that g L, and Then g is Skorohod integrable with respect to Z and E g s δz(s E D x1,x g Hdx 1 dx <. (34 [ ] cst. E g H + E D x1,x g Hdx 1 dx. (35 Proof: We use Meyer s inequality for the double Skorohod integral (see [8], pag. 3 and 18

19 we obtain E g s δz(s = cst. [ cst. E +E + = cst. [ EH(H 1 I(g(y 1, y dy 1 dy (D x1,x I(g(y 1, y dx 1 dx dy 1 dy ] dx 1 dx ( [ E g H + E g(ug(v u v H dvdu ] D x1,x g(ud x1,x g(v u v H dvdu D x1,x g Hdx 1 dx ]. Corollary 3 If g L (Ω; H be such that g L, and Then g is Skorohod integrable with respect to Z and E D x1,x g H dx 1dx <. (36 E g s δz(s cst. g (37 where g = [E g H + E D x1,x g H dx 1dx ]. Example 1 The Rosenblatt process Z is Skorohod integrable with respect to Z and E g s δz(s cst. R(u, v u v H dudv. Proof: We treat the two terms in the right side of (35. Clearly We have, for every x 1, x [, T ], E Z H = cst. R(u, v u v H dudv. u D x1,x Z(u = d(h1 [,u] (x 1, x x 1 x (u, x 1 KH (u, x du 19

20 and then (note also that KH t (t, s is positive and therefore we omitt the absolute value at a certain point = = E u x 1 x = cst. D x1,x g H dx 1dx dx 1 dx x 1 x x 1 x u v H dudv (u, x 1 KH (u, x du u v H dudv u R(u, v u v H dudv. ( u v x 1 x (v, x 1 KH (v, x dv (u, x 1 K (v, x 1 dx 1 We finish the section by a result on the continuity of the indefinite Skorohod integral process. This shows that the indefinite keeps the same order of Hölder regularity as the Rosenblatt process. Proposition 4 Let g L,p such that sup g r,p. r ( Then the indefinite Skorohod integral process X t = t g sδz(s, t [, T ] continuous version of order δ < H. admits a Hölder Proof: Then We can write X t X s = s + + ( s y 1 y ( t s ( := J 1 + J + J 3. s y (u, y 1 KH (u, y du (u, y 1 KH (u, y du (u, y 1 KH (u, y du E X t X s p c(pe (J p 1 + J p + J p 3. db(y 1 db(y db(y 1 db(y db(y 1 db(y

21 By Meyer s inequality ([8], pag. 3 E J 1 p t ( t c(p E s s +c(pe s s = c(p, H E s s g u y 1 y [ D x1,x (u, y 1 KH (u, y du g u y 1 y g(ug(v u v H p dy 1 dy p (u, y 1 KH (u, y du +c(p, HE dx 1 dx D x1,x g u D x1,x g v u v H dvdu s s c(p, H sup g r p t p,p u v H dvdu r = c(p, H sup r s s g r p,p (t sph. ] dx 1 dx dy 1 dy In a similar way, we can find the same bound for the terms J and J 3 (see also [4], proof of Proposition 1. The conclusion will following by the Kolmogorov s continuity criterium. p p 7 The relation between the pathwise and the Skorohod integrals Let g a stochastic process. Recall that its forward integral with respect to Z is the limit ucp as ε of I (ε, g, dz = 1 ε where the kernel f s is given by g s (Z(s + ε Z(sds = 1 ε f s (x, y = d(h1 [,s] (x, y x y g s δ ( (f s+ε (, f s (, ds (38 (u, x KH (u, ydu. (39 We will need a formula by [8]: if F D,, u L ([, T ] Ω such that for every s, u(, s Dom(δ, then F u Dom(δ ( and F δ ( (u = δ ( (F u + D α F δ(u(, αdα D ( α,βf u(α, βdαdβ. (4 1

22 We apply relation (4 to (38 and we obtain I (ε, g, dz = 1 ε + ε 1 ε δ ( (g s (f s+ε (, f s (, ds D α g s δ (f s+ε (, α f s (, α dαds D ( α,β g s (f s+ε (β, α f s (β, α dβdαds. (41 We can already observe, besides the first divergence type term, the appearance of two trace terms. Recall that in the fbm case the corresponding term I (ε, g, db H can be decomposed in a divergence term plus a only a trace term. Definition We say that a stochastic process g L 1, admits a trace of order 1 if 1 ε D α g s δ (f s+ε (, α f s (, α dαds (4 converges in probability as ε. The limit will be denoted by T r (1 (D (1 g. We say that a stochastic process g L, admits a trace of order if 1 ε D ( α,β g s (f s+ε (β, α f s (β, α dβdαds (43 converges in probability as ε. The limit will be denoted by T r ( (D ( g. We have the following relation between the divergence and the pathwise integral. Theorem Let g L, such that E g H + E D x1,x g H dx 1dx <. Assume that g has traces of order 1 and. Then g is forward integrable with respect to Z and it holds Proof: g s d Z(s = g s δz(s + T r (1 (D (1 g T r ( (D ( g. (44 By (41 and Definition, it suffices to show that the term converges to A ɛ = 1 ε δ ( (g s (f s+ε (, f s (, ds g s δz(s

23 in L (Ω as ε. We can write, by Fubini, A ε = 1 ε = = where we denoted by ds db(y 1 db(y g s (f s+ε (y 1, y f s (y 1, y db(y 1 db(y g ε (u KH y 1 y I(g ε (y 1, y db(y 1 db(y = g ε (u = 1 ε By using (37, it is sufficient to check that and We will show that u u ε g ε ε g in L (Ω; H (u, y 1 KH (u, y du g ε sδz(s E D x1,x (g ε g Hdx 1 dx ε. g s ds. (45 g ε H c(h g H (46 and D x1,x g ε H dx 1dx c(h D x1,x g H dx 1dx. (47 The bound (46 has been proved in [5], proof of Proposition 3, Step 1. Concerning the bound (47, we can write = c(h D x1,x g ε H dx 1dx c(h 1 ε dx 1 dx dx 1 dx D x1,x g ε u D x1,x g ε v u v H dudv dudv u v H v ε c(h dx 1 dx dsds D x1,x g s D x1,x g s It follows from [5], proof of Proposition 3, Step 1, that 1 ε +ε +ε s s u dsds D x1,x g s D x1,x g s u ε +ε +ε ( 1 ε u v H dudv c(h s s H 3 s s u v H dudv.

24 and thus (47 follows. Now we can finish the proof proceeding as in [5], proof of Proposition 3, Step 3. Consider a sequence g n of simple processes of the form g n = n 1 i= F i1 (ti,t i+1 ] with F i S and t i [, T ] such that g n g in L (Ω when n (the existence of a such sequence follows easily by the densite of E in H. Then by (37 we have that gs n δz(s n g s δz(s Denote by g n,ε the approximation process of the form (45 associated to g n. We can write, for any ε > and n 1, ( E gsδz(s ε T g s δz(s 3 E gsδz(s ε gs n,ε δz(s +E gs n,ε δz(s gs n δz(s + E gs n δz(s g s δz(s. By (46 and (47 it follows that for n large enough and for any δ > ( E gsδz(s ε T g s δz(s 3 E gs n,ε δz(s g n s δz(s and we can conclude by taking ε. + δ 8 On the Itô formula in the Skorohod sense We study Itô s formula for the Rosenblatt process in the divergence sense. As we mentioned before, the Gaussian nature of the integrator process is essential in the framework of the divergence calculus and this fact can be entirely observed here. We are actually able to prove Skorohod Itô s formula only in two particular cases; but more relevant than these formulas, which are not easily tractable, is the fact that one can observe from the computations contained here that the standard method to obtain divergence type change of variables formulas (see e.g. [7] does not work here, in the sense that one cannot hope to obtain Itô s formulas that stop at f. We will deduce the Skorohod Itô formula by using the pathwise Itô formula. Recall that for any function f C (R f(z(t = f( + = f( + f (Z(sd Z(s provided that the above terms exist. f (Z(sδZ(s + T r (1 (D (1 f (Z(s T r ( (D ( f (Z(s 4

25 8.1 The trace of order 1 Recall that where B ε = 1 ε = 1 ε T r (1 (D (1 f (Z(s = ucp lim ε B ε ds ds dαd α f (Z(sδ (f s+ε (, α s s (, α dαf (Z(sD α Z(sδ (f s+ε (, α f s (, α. The Malliavin derivative of Z(s is given by ( ( s s D α Z(s = d(h1 [,s] (α α y 1 (u, α KH (u, y 1du db(y 1 (48 Thus B ε = ε d(h dsf (Z(s δ (f s+ε (, α f s (, α dα ( s s α y 1 (u, α KH (u, y 1du db(y 1 where f s is given by (39. By using the integration by parts formula (31 it holds that B ε = ε d(h dsf (Z(s [ δ (f s+ε (, α f s (, α dα α y 1 (u, α KH (u, y 1du ] db(y 1 + ε d(h dsf (Z(s dα [ s ] s (u, α KH α y 1 (u, y 1du(f s+ε (y 1, α f s (y 1, α dy 1 := B 1 ε + B ε. 5

26 We regard first the term Bε because it can be treated in the same manner for any function f. We can write Bε = ε d(h dsf (Z(s [ dy 1 α y 1 +ε 1 [,s+ε] (y 1, α dα (u, α KH (u, y 1du α y 1 1 [,s] (y 1, α α y 1 = ε d(h dsf (Z(s = A(H (v, α KH (v, y 1dv ] (v, α KH (v, y 1dv ( s+ε u v du dv dv du u v H 1 ε with A(H = H (H 1d(H and we have B ε = A(H dv +A(H dv s (v ε u f (Z(sds ( du u v H 1 v ε v ε du u v H ( 1 ε Therefore we have the convergence in L 1 (Ω as ε B ε A(H u (v ε u (u, α KH (v, αdα f (Z(sds f (Z(sds. (49 f (Z(v u v H dvdu = A(H f (Z(uu H 1 du (5 H 1 since the first summand in (49 converges to the limit and the second one goes to zero by the dominated convergence theorem. The study of the term B 1 is rather difficult to be done in general. We will restrict ourselves to its study in some particular cases. The case f(x = x : We have Bε 1 = 4 ε d(h ds α y 1 dα db(y 1 db(y (u, α KH (u, y 1du(f s+ε (y, α f s (y, α 6

27 and by Fubini we get B 1 ε = 4d(H db(y 1 db(y ( ( 1 v u v ds ε (v ε u = 4d(H H (H 1 y dv and then one can prove that du KH y 1 B 1 ε 4d(H H (H 1 y dv du KH y 1 y dv du KH y 1 (u, α KH (v, αdα db(y 1 db(y (u, y 1 KH (u, y u v H db(y 1 db(y (u, y 1 KH (u, y ( 1 v ds ε (v ε u (u, y 1 KH (u, y u v H. (51 The case f(x = x 3 : It holds by calculating first the integral dα Bε 1 = 1 ε d(h H (H 1 Z(sds [ +ε db(y 1 db(y du y 1 db(y 1 db(y du y 1 +ε y dv KH dv KH y (u, y 1 KH (v, y u v H ] (u, y 1 KH (v, y u v H and the integration by parts formula for the double Skorohod integral gives where Bε 1,1 = 1 ε d(h H (H 1 ds [ +ε db(y 1 db(y Z(s Bε 1 = Bε 1,1 + Bε 1, + Bε 1,3 y 1 du db(y 1 db(y Z(s du y 1 7 +ε y dv KH dv KH y (u, y 1 KH (v, y u v H ] (u, y 1 KH (v, y u v H,

28 Bε 1, = 48 ε d(h3 H (H 1 ( ds dα du KH α y 1 (u, α KH (u, y 1 [ +ε db(y db(y α α du du +ε y dv KH dv KH y db(y 1 (u, α KH (v, y u v H ] (u, α KH (v, y u v H and Bε 1,3 = 4 ε d(h3 H (H 1 [ +ε dy 1 dy y 1 du +ε y y 1 du dv KH dy 1 dy dv KH y ds y 1 y (u, y 1 KH (u, y du (u, y 1 KH (v, y u v H y 1 y (u, y 1 KH (u, y du (u, y 1 KH (v, y u v H ] Now, B 1,1 ε = 1d(H H (H 1 y dv du KH y 1 db(y 1 db(y (u, y 1 KH (v, y u v H and we have the convergence as in the proof of Theorem ( 1 v Z(sds ε (v ε u B 1,1 ε 1d(H H (H 1 y dv duz v y 1 db(y 1 db(y (u, y 1 KH (v, y u v H (5 To treat the term B 1, ε one needs to use again the integration by parts formula (31 and one 8

29 obtains B 1, ε 48d(H 3 (H (H 1 y dv Concerning B 1,3 ε B 1,3 ε y 1 du db(y 1 db(y du u u H u v H KH y 48d(H 3 (H (H 1 3 dv we similarly have 4d(H 3 (H (H 1 3 (u, y 1 KH (v, y du du u v H u v H u u H. dv du du u v H u v H u u H. (53 8. The trace of order Recall that where C ε = 1 ε ( T r ( D ( f (Z(s = ucp lim C ε ε ds D α,β f (Z(s (f s+ε (α, β f s (α, β dαdβ = 1 ds (f s+ε (α, β f s (α, β dαdβ ε [ f (Z(sD α,β Z(s + f (Z(sD α Z(sD β Z(s ] := C 1 ε + C ε. We can write C ε = ε d(h f (Z(sds du +ε s dv = ε d(h (H (H 1 f (Z(sds = d(h (H (H 1 dv ( u v du +ε du u v H ( 1 ε s (u, α KH (v, αdα dv u v H (v ε u f (Z(sds and then clearly in L 1 (Ω as ε. C 1 ε d(h (H (H 1 H 1 f (Z(vv H 1 dv. (54 9

30 C ε = 1 ε The term denoted by C ε can be handled in the following way: dsf (Z(s = 4 ε d(h dsf (Z(s ( db(y 1 α y 1 D α Z(sD β Z(s (f s+ε (α, β f s (α, β dαdβ dαdβ (f s+ε (α, β f s (α, β ( (u, α KH (u, y 1du db(y β y (v, β KH (v, y dv The case f(x = x 3. In this case Cε = 4 ε d(h + 4 ε ds db(y 1 db(y ds dy 1 α y 1 K dαdβ (f s+ε (α, β f s (α, β α y 1 (u, α KH dαdβ (f s+ε (α, β f s (α, β (u, α KH (u, y 1du β y (u, y 1du β y (v, β KH (v, y dv and by the same type of calculations as above we can prove that as ε C ε 4d(H 3 (H (H 1 u u H v u H K (u, y 1 KH (v, y u +4d(H 3 (H (H 1 3 du (v, β KH (v, y dv u u db(y 1 db(y du du dv (55 y 1 y 1 y u dudv u u H v u H u v H. (56 We can summarize Theorem 3 We have + Z(t = 4(H 1 H + 1 Z(sδZ(s + t H u db(y 1 db(y dv y du KH y 1 (u, y 1 KH (u, y u v H. 3

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