Itô formula for the infinite-dimensional fractional Brownian motion

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1 Itô formula for the infinite-dimensional fractional Brownian motion Ciprian A. Tudor SAMOS-MATISSE Université de Panthéon-Sorbonne Paris 1 9, rue de Tolbiac, Paris France. January 13, 25 Abstract We introduce the stochastic integration with respect to the infinite-dimensional fractional Brownian motion. Using the techniques of the anticipating stochastic calculus, we derive an Itô formula for Hurst parameter bigger than Introduction The fractional Brownian motion (fbm B h starting from zero, with covariance = (B h t t [,1] is a centered Gaussian process, R(t, s = 1 2 (t2h + s 2h t s 2h for every s, t [, 1]. The parameter h belongs to (, 1 and it is called Hurst parameter. If h = 1 2 the associated process is the classical Brownian motion. The process (Bh t t [,1] has stationary increments and it is self similar, that is, Bαt h and α h Bt h have the same distribution for all α >. These properties make the fbm a candidate as a model in different applications (like network traffic analysis or mathematical finance. Therefore a stochastic calculus with respect to the fractional Brownian motion was needed. Let us briefly recall the two principal directions considered in the fractional stochastic integration. a first approach is the anticipating (Skorohod stochastic calculus and the white noise theory. This approach has been used in e.g. [2], [5] or [6] among others. An Itô s formula involving the Skorohod (divergence integral has been proved. Note that for small Hurst parameters one need an extended divergence integral (see [4]. the pathwise integration theory has been considered in e.g. [8], [18] or [1]. An Itô formula has been proved also in this case and stochastic equations in the Stratonovich sense driven by the fbm has been considered. Again, a generalized integral was needed for small Hurst parameters. 1

2 A natural extension of this problems is to develop a stochastic calculus with respect to a Hilbert space-valued fbm. Recently, the infinite dimensional fbm has been considered by several authors as a driving noise in the study of stochastic evolution equations. We refer, among others, to [7], [9] or [16]. Note that the stochastic integrals with respect to the infinite dimensional fbm appearing in these papers are only Wiener integrals. Our aim is to introduce the stochastic integration of non-deterministic integrands with respect to a Hilbert-valued fbm and to obtain an Itô formula in the Skorohod sense. Our work is motivated, on one side, by the recent development of the stochastic integration with respect to Gaussian (and even more general processes and on the other side, by practical aspects. For example, we believe that a such stochastic integration could open the door to a further study of the connection between the ordinary and stochastic fractional calculus. Let us recall that, in the case of the infinite dimensional Wiener process, the last term appearing in the Itô formula (the trace term, see e.g. [15] corresponds to the Kolmogorov equation associated to the infinite dimensional Ornstein-Uhlenbech process. See Section 9 in [15] for details. This provides an intimated connection between the theory of the infinite dimensional diffusions and the theory of partial differential equations with concrete practical applications such that e.g. the wavefront propagation. In the case of the Hilbert space-valued fbm the last term is the one we could expect (see Section 4 below and this seems to be related to the fractional Kolmogorov equation which is a reaction-diffusion equation with a fractional diffusion operator (see [3] and [12] for the definition and applications. We plan a separate study of all this relations; we just observe that trace class operators appear also in the study of the stochastic evolution equations with fbm (see Theorem 1 in [16]. Our approach extends, on one hand, the results of [1] in the case h = 1 2 and, on the other hand, the results of [2] in the one-dimensional fbm case. We mention that the passage from the one-dimensional to the infinite dimensional case needs new techniques; for example the methods used in [2] to prove the Itô s formula are not directly applicable to the Hilbert-valued situation; the difficulty comes from the fact that the proof in [2] is based on a specific one-dimensional duality relation between the Skorohod integral and the Malliavin derivative. We prove here that, in the Wiener case, the approach of [15] to define stochastic integrals on Hilbert spaces and the method of [1] lead to the same integral and we take advantage from this fact. The method that we use to prove the Itô s formula for h > 1 2 is the classical Taylor expansion. Our paper is organized as follows: Section 2 is devoted to recall the basic notions of the stochastic calculus of variations with respect to the Wiener process and fractional Brownian motion. In Section 3 we present the construction of the adapted and non-adapted infinitedimensional stochastic analysis and we study the relation between different approaches. In Section 4 we introduce a fbm B h in a Hilbert space and we obtain an Itô formula. 2

3 2 Preliminaries 2.1 Malliavin Calculus Let T = [, 1] the unit interval and (W t t T the standard Wiener process on the canonical Wiener space (Ω, F, P. We will denote by S the class of Brownian functionals of the form F = f(w t1,, W tn (1 where f : R n R is an infinitely differentiable function such that f and all its derivatives are bounded and t 1,, t n T. The elements of S are called smooth random variables and form a dense subspace of L 2 (Ω. The Malliavin derivative of a smooth functional F of the form (1 is the stochastic process {D t F ; t T } given by D t F = n i=1 f x i (W t1,, W tn 1 [,ti ](t, t T More generally, we can introduce the k-th derivative of F S by D (k t 1, t k F = D t1 D t2 D tk F. Consider now V a real and separable Hilbert space and S V the class of the smooth V -valued random variables that can be written as F = n j=1 F jv j where F j S and v j V. We shall introduce the derivative of F S V as D s F = n D s F j v j, s T. (2 j=1 This operator is closeable from L p (Ω; V into L p (T Ω; V for any p 1 and it can be extended to the completion of S V, denoted D k,p (V, with respect to the norm k F p k,p,v = E F p V + E D (j F p L 2 (T j ;V. The adjoint of D, called the Skorohod integral and denoted by δ, is characterized by the duality relationship E δ(u, F V = E DF, u L 2 (T V for all F S V, if u belongs to Dom(δ (the domain of the operator δ, where j=1 Dom(δ = {u L 2 (T Ω; V / E DF, u L 2 (T V C F L 2 (Ω;V }. We will need the following property of the Malliavin derivation in Hilbert spaces (see [13] (D α F (x = D α (F (x (3 if U, V are two Hilbert spaces, F is a random variable taking values in L(U, V, x U and α T. 3

4 2.2 Fractional Brownian motion Let B = (B t t T be the fractional Brownian motion (fbm with Hurst parameter h (, 1. We will omit in this section the superindex h. We know that B admits a representation as Wiener integral of the form B t = t K(t, sdw s, where W = (W t t T is a Wiener process, and K(t, s is the kernel (see [2, 5] ( K(t, s = c h (t s h s h 1 t 2 F1, s ( c h being a constant and F 1 (z = c 1 h 2 h ( z 1 θ h (θ + 1 h 1 2 dθ. This kernel satisfies the condition K t (t, s = c h(h 1 2 (s t 1 2 h (t s h 3 2. (4 We consider the canonical Hilbert space of the fbm H as the closure of the linear space generated by the function {1 [,t], t T } with respect to the scalar product 1 [,t], 1 [,s] H = R(t, s. Then the mapping 1 [,t] B t gives an isometry between H and the first chaos generated by {B t, t T } and B(φ denotes the image of a element φ H. We recall that, if φ, χ H are such that T T φ(s χ(t t s 2h 2 dsdt <, their scalar product in H is given, when h > 1 2, by φ, χ H = h(2h 1 φ(sχ(t t s 2h 2 dsdt. (5 We can introduce a derivation and a Skorohod integration with respect to B. For a smooth H-valued functionals F = f(b(ϕ 1,, B(ϕ n with n 1, f Cb (Rn and ϕ 1,, ϕ n H we put D B (F = n j=1 f x j (B(ϕ 1,, B(ϕ n ϕ j. and D B F will be closable from L p (Ω into L p (Ω; H for any p 1. Therefore, we can extend D B to the closure of smooth functionals D k,p B with respect to the norm F p B,k,p = E F p + k (D B (j F p L p (Ω;H j. Consider the adjoint δ B of D B. Its domain is the class of u L 2 (Ω; H such that and δ B is the element of L 2 (Ω given by j=1 E D B F, u H C F 2 for all F D 1,2 B E(δ B (uf = E D B F, u H for every F smooth. The following relations will relate the derivation and the Skorohod integration with respect to W and B. 4

5 (i D k,p = D k,p B, and K 1 DB F = DF, for any F D k,p. (ii Dom(δ B = (K1 1 (Dom(δ, and δ B (u = δ(k1 u for any H-valued random variable u in Dom(δ B, where (K 1ϕ(s = s ϕ(r K (r, sdr. (6 r We have the integration by parts formula, provided that all terms have sense, F δ B (u = δ B (F u + D B F, u H. (7 Throughout this paper, we denote by HS the Hilbert-Schmidt norm. 3 Infinite-dimensional stochastic analysis Consider T the unit interval and let U be a real separable Hilbert space. We consider Q a nuclear, self-adjoint and positive operator on U (Q L 1 (U, Q = Q >. It is wellknown that Q admits a sequence (λ j of eigenvalues with < λ j and λ j <. Moreover, the corresponding eigenvectors (e j form an orthonormal basis in U. A process (X t t T with values in U is called Gaussian if for every t 1,, t n T and u 1,, u n U the real random variable n j=1 u j, X tj U has a normal distribution. We define the expectation m X of X as m X : T U, m X (t = E[X t ] and the covariance C X : T 2 L 1 (U by, for every s, t T, C X (t, su, v U = E[ X t m X (t, v U X s m X (s, u U ]. Definition 1 We call (W t t T a Q-Wiener process for the filtration F s if W is a U -valued process and the following properties hold: i W = and W has continuous trajectories. ii For every s < t, W t W s is independent of F s and W t W s N(, (t sq. We have equivalent definitions for a Q-Wiener process(see [17]. Theorem 1 Let (W t t T a U-valued process such that W = and W has continuous trajectories. The following are equivalent: i W t is a Q-Wiener process. ii W t is a centered Gaussian process with covariance C W (t, s = min(t, sq. iii There exist real and independent Brownian motions {(β j (t t T } such that W t = λj β j (te j. (8 5

6 Let H be another real separable Hilbert space with (h k k an orthonormal system in H and put U = Q 1 2 (U U. We endow U with the norm u = Q 1 2 (u U. Then (U, is a separable Hilbert space and ( λ j e j ( is a orthonormal basis in U. 1 For a L 2 (U ; H-valued, adapted process with E Φ s 2 HS ds <, the Itô stochastic integral of Φ with respect to W is defined as being the H-valued process (I Φ (t t T given by I Φ (t = λj Ij Φ (t (9 with I Φ j (t = k 1 ( t Φ s e j, h k dβ j (s h k and the two series above are convergent in L 2 (Ω; H, uniformly on T. Note that I Φ is a martingale on H (see [15] or [17] for the definition. We will also recall some notions of the non-adapted stochastic calculus with respect to W following [1]. A different construction, based on chaos expansion, was given in [11]. A Malliavin type derivative operator and a Skorohod integral with respect to the infinitedimensional Brownian motion (W t t T has been introduced in [1] as follows. Denote, for h L 2 (T ; U, by W(h = h(sdw s L 2 (Ω the Wiener integral h(s, dw s U (which is equal to λj h(s, e j U dβ j (s if W is given by (8 and consider S H the subspace of L 2 (Ω; H of smooth functionals F = m f p (W(h p1,, W(h pnp k p p=1 where m, n 1,, n m N, k 1,, k m H, f p Cb (Rn p for all p = 1,, m and h pi L 2 (T ; U for all i = 1,, n p. Then the derivative of F is the L 2 (U ; H-valued process {D t F ; t T } with D t F = n m p p=1 i=1 f p x i (W(h p1,, W(h pnp k p h pi (t. The operator D can be extended to the closure of S H with respect to the norm F 2 1,2 = E F 2 H + E D s F 2 L 2 (U ;H ds. We define the Skorohod integral δ(φ of a process Φ L 2 (T Ω; L 2 (U ; H as being the H-valued random variable δ(φ characterized by the following duality E F, δ(φ H = E 6 D s F, Φ s HS ds (1

7 and this operator is well-defined on the set of processes Φ L 2 (T Ω; L 2 (U ; H such that E D s F, Φ s HS ds C F L 2 (Ω;H. When W is a real one-dimensional Brownian motion, the operators D and δ coincide with the ones defined in Section 2.1. We denote by L 1,2,Q (H the Hilbert space of processes Φ belonging to L 2 (T Ω; L 2 (U ; H with the norm Φ 2 1,2,Q = E Φ s 2 HSds + E By Prop of [1], this space is included in the domain of δ. D α Φ s 2 HSdαds. Remark 1 Observe that L 1,2,Q (H can be defined also as the class of stochastic processes (Φ s s T with values in L 2 (U, H such that Φ s D 1,2 (L 2 (U, H for every s T and λ j E Φ s e j 2 Hds < (11 and λ j E (D α Φ s e j 2 Hdαds <. (12 The next result shows that the Skorohod integral of [1] can be defined, for enough regular non-adapted integrands, in the same way as the Itô integral (9. Proposition 1 Let Φ L 1,2,Q (H.The following properties hold: i For every j 1, the following series converge in L 2 (Ω; H Sj Φ = ( Φ s e j, h k H dβ j (s h k and S Φ = λj Sj Φ ; (13 k 1 ii S Φ coincides with the Skorohod integral δ(φ given by (1. Proof: i We note first that the real Skorohod integral exists. Indeed, E Φ s e j, h k 2 Hds t Φ s e j 2 Hds 1 λ j E λ j Φ s e j 2 Hds < and E = E (D α Φ s e j, h k H 2 dαds = E (D α Φ s e j, h k 2 Hdαds E D α (Φ s e j, h k 2 Hdαds (D α Φ s e j 2 Hdαds <. 7

8 We will use the notation Φ s e j, h k H = Φ j,k (s. The series (13 converges in L 2 (Ω because E k k E Φ j,k (sdβ j (sh k 2 = k Φ j,k (s 2 ds + k E ( t 2 E Φ j,k (sdβ j (s (D α Φ j,k (s 2 dαds. and this is finite since and E k 1 E k 1 Φ j,k (s 2 ds = E (D α Φ j,k (s 2 dαds = E Φ s e j 2 Hds < (D α Φ s e j 2 Hdαds <. Concerning the second sum in (13, we observe first that, for j l, E[Sj Φ(tSΦ l (t] = and then, due to the nuclearity of Q E λj Sj Φ (t 2 H = λ j E(Sj Φ (t 2 Φ 2 L <. 1,2,Q j j ii The next step is to prove that, for Φ L 1,2,Q and F S H, S Φ and F satisfy the duality relation E S Φ, F H = E Φ s, D s F HS ds. (14 For simplicity, take F = f(w(hu, with h L 2 (T ; U and u H. We have Therefore it holds D s F, Φ s L2 (U,H = j D t F = f (W(hh t u. (15 λ j (D s F (e j, Φ s e j H = j λ j f (W(h(h s u(e j, Φ s e j H = j λ j f (W(h h s, e j U u, Φ s e j H. On the other hand E S Φ, F H = j,k 1 = j,k 1 ( 1 λj E Φ s e j, h k H dβ j (s f(w(h u, h k H λj E ( = ( 1 λj E Dsf(W(h Φ j s e j, h k ds u, h k H Dsf(W(h Φ j s e j, u H ds, 8

9 where D j denotes the Malliavin derivative with respect to the Brownian motion β j. Thus the duality follows observing that D j sf(w (h = f (W(hD j s(w(h = f (W(h λ j h s, e j U. The relation (14, implies that Φ belongs to Dom(δ and thus L 1,2,Q Dom(δ and S Φ coincides with δ(φ. 4 Infinite-dimensional fbm and Itô formula With the notation of Section 3 we introduce a fbm B h on U as follows. Definition 2 We say that the U-valued process (B h t t T is an infinite-dimensional fractional Brownian motion (or a Q fbm if B h is a centered Gaussian process with covariance C B h(t, s = R(t, sq. Proposition 2 B h is a Q fbm if and only if there exists a sequence (β h j of real and independent fbm such that B h t = λj β h j (te j (16 where the series converges in L 2 (Ω; U. Proof: We refer to [7] or [9] for the fact that λj βj h(te j is a fbm in the sense of Definition 2. Conversely, let B h be a centered Gaussian process on U with covariance R(t, sq. Put βj h(t = 1 B h λj t, e j U. Then βj h is a Gaussian process since n α l βj h (t l = l=1 n B h α l t l, e j λj l=1 for every α 1,, α n R and t 1,, t n T. Moreover, for every i, j 1,s, t T, we have ( E βi h (sβj h (t = 1 E ( B h s, e i U B h t, e j U = R(t, sδ ij λi λ j and it implies that E ( βi h(sβh i (t = R(t, s and for i j, the random variables βi h(s and βj h (t are uncorrelated, thus independent. We finish the proof by noting that the sum λj βj h(te j converges in L 2 (Ω; U. Remark 2 Note that the process B h always has a continuous version (see [15], Prop

10 Remark 3 When the covariance operator Q is not nuclear we can still introduce an infinite dimensional fbm with covariance Q. As before, let U real separable Hilbert space and Q L(U, Q = Q > (Q is not necessary nuclear, in particular Q may be the identity operator. Let U 1 U be an other real and separable Hilbert space and (g j j an orthonormal basis in U such that the mapping J : (U, (U 1, 1 is a Hilbert-Schmidt operator, i.e. Jg j 2 1 <. Consider the operator Q 1 = JJ : U 1 U 1 which is nuclear, positive and self-adjoint and let (β h j be real and independent fbm with h (, 1. Then the U 1 - valued process B h t = (Jg j β h j (t (17 is an Q 1 fbm, provided that the series (17 converges in L 2 (Ω. Let us fix h (, 1 and consider D 1,2,Q K (H the class of processes Φ with values in L 2 (U ; H such that λ j E Φe j H 2 H < and λ j E (DΦe j H 2 H H <. Definition 3 If B h is an infinite dimensional fbm in the form (16 and Φ is a process in the space D 1,2,Q (H, then we define K Φ s db h s = λj T j (Φ (18 where T j (Φ = k 1 ( t Φ s e j, h k dβj h h k. (19 As in the proof of Proposition 1, one can show that the sums (18 and (19 are finite for Φ D 1,2,Q K (H. We prove now the Itô formula for the U-valued fbm with nuclear covariance Q. Theorem 2 Let (B h t t T a U-valued fbm with h ( 1 2, 1 and let F : U R, F C3 b (U such that F, F are uniformly continuous. Then it holds F (B h t = F ( + where the last term is defined as t t F (B h s s 2H 1 ds := F (B h s db h s + H t F (B h s s 2H 1 ds t λ j F (B h s (e j (e j s 2H 1 ds. 1

11 Proof: Let π : = t < t 1 < < t n = t denote a partition of the interval [, t]. We write the following version of the Taylor formula in the differential calculus F (B h t = F ( + F (B h t i (B h t i+1 i= + 1 F ( B h t 2 i (B h t i+1 (B h t i+1 i= where B h t i is located between B h t i and B h t i+1. Using the definition (16 of B h and the linearity of F (x, we obtain F (B h t = F ( + F (B h t i λj (βj h (t i+1 βj h (t i e j i= + 1 F ( B h t 2 i (B h t i+1 (B h t i+1 i= = F ( + λj (βj h (t i+1 βj h (t i F (B h t i (e j i= + 1 F ( B h t 2 i (B h t i+1 (B h t i+1. i= By the integration by parts formula (7, it holds F (B h t = F ( + + λj λj i= ( δ h,j (1 (ti,t i+1 ]( F (B h t i (e j D. h,j F (B h t i (e j, 1 (ti,t i+1 ]( H i= + 1 F ( B h t 2 i (B h t i+1 (B h t i+1 i= where δ h,j and D h,j denotes respectively the Skorohod integral and the Malliavin derivative with respect to the real fractional Brownian motion βj h. Since, by (3 Dα h,j F (B h t i (e j = F (B h t i Dα h,j B h t i (e j, Dα h,j B h t i = λk Dα h,j βk h (t ie k = λ j 1 [,ti ](αe j k 1 11

12 we will have F (B h t = F ( + + λ j λj i= ( δ h,j 1 (ti,t i+1 ]( F (B h t i (e j 1 (,ti ], 1 (ti,t i+1 ] H F (B h t i (e j (e j i= + 1 F ( B h t 2 i (B h t i+1 (B h t i+1. i= Notice that, if R(t := R(t, t, 1 (,ti ], 1 (ti,t i+1 ] H = R(t i, t i+1 R(t i = 1 2 (t2h i+1 t 2h i 1 2 (t i+1 t i 2h and, therefore, the last sum becomes F (B h t = F ( λ j λj i= ( δ h,j 1 (ti,t i+1 ]( F (B h t i (e j F (B h t i (e j (e j ( R ti+1 R ti i= F (B h t i (e j (e j (t i+1 t i 2h λ j i= + 1 F ( B h t 2 i (B h t i+1 (B h t i+1 i= := F ( + T 1 + T 2 + T 3 + T 4. Step 1: We regard the stochastic integral term T 1 = λj i= ( δ h,j 1 (ti,t i+1 ]( F (B h t i (e j (2 and we will show its convergence to ( λj δ h,j F (B ḥ (e j 1 [,t] ( = t λj F (B ḥ (e j dβj h (s in L 2 (Ω as π. First, we prove the L 2 (Ω H-convergence to, as π of the sum ( 1 (ti,t i+1 ](s F (B h t i F (B h s (e j. i= 12

13 We compute E i= ( 1 (ti,t i+1 ](s F (B h t i F (B h s (e j 2 H = Since E i= ( 1 (ti,t i+1 ](s F (B h t i F (B h s (e j, l= ( 1 (tl,t l+1 ](s F (B h t l F (B h s (e j H. ( F (B h t i F (B h s (e j F B h t i B h s U using the form of the scalar product (5, the last sum will be lesser than F 2 sup a b π B h b Bh a 2 U 1 (ti,t i+1 ], 1 (tl,t l+1 ] H and that goes to using the continuity of B h and i,l= 1 (t i,t i+1 ], 1 (tl,t l+1 ] H = t 2h. Now, let s regard the convergence of the derivative of the sum i= i,l= ( 1 (ti,t i+1 ](s F (B h t i F (B h s (e j in L 2 (Ω H H-convergence to, as π. We have, by (3 D h,j α Therefore 1 (ti,t i+1 ](sf (B h t i (e j = λ j 1 (ti,t i+1 ](s1 (,ti (αf (B h t i (e j (e j, ( i= D h,j α i= ( F (B h s (e j = λ j 1 (,s (αf (B h s (e j (e j. ( ( E D h,j 1 (ti,t i+1 ](s F (B h t i F (B h s (e j 2 H H 2λ j E i= i= 1 (ti,t i+1 ](s1 (,ti (α ( F (B h t i F (B h s (e j (e j 2 H H ( +2λ j E 1 (ti,t i+1 ](s1 (ti,s(α F (B h s (e j (e j 2 H H = A 1 + A 2. i= The first summand A 1 goes to from the following bounds A 1 = 2λ j F 2 sup B h b Bh a 2 U a b π 2λ j t 2h F 2 sup a b π 1 (ti,t i+1 ], 1 (tl,t l+1 ] H 1 (,ti ], 1 (,tl ] H i,l= B h b Bh a 2 U 13

14 since 1 (,ti ], 1 (,tl ] H 1 and i,l= 1 (t i,t i+1 ], 1 (tl,t l+1 ] H = t 2h. For the second summand A 2, using the definition of the scalar product in H H, we have ( 2 A 2 2λ j 1 (ti,t i+1 ], 1 (tl,t l+1 ] 2 H 2λ j t i+1 t i 2h. i,l= i= Step 2: Clearly the term T 2 converges in L 1 (Ω to t F (B h s dr s = t λ j F (B h s (e j (e j dr s. Step 3: Recall that h > 1 2. It is easy to observe that T 3 converges to since E T (T rq F (t i+1 t i 2h 1 2 (T rq F π 2h 1. i= We finally study the term T 4. We can write, since by hypothesis for every x U, F (x is a bounded continuous operator in L 2 (U 2 ; R, E T F E B h t i+1 2 U This finishes the proof. i= 1 2 F T r(q (t i+1 t i 2h π. Remark 4 The proof of Theorem 2 can be also applied to the one-dimensional case and it is an alternative proof to the one given in [2]. The study of the indefinite integral process (continuity of the paths, Itô formula can be done in the Hilbert space-valued situation without difficulty, following the lines of the one-dimensional case. Recently, in [4], the authors extended the divergence integral with respect to fbm for any parameter H (, 1. We think that their approach can be used in the infinite-dimensional context. i= References [1] E. Alòs, J. Léon and D. Nualart, Stratonovich calculus for fractional Brownian motion with Hurst parameter less than 1 2, Taiwanese Journal of Math., 4 (21, [2] E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes, Annals of Probability, 29(2 (21,

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