/4 On the -variation of the divergence integral w.r.t. fbm with urst parameter < 2 EL ASSAN ESSAKY joint work with : David Nualart Cadi Ayyad University Poly-disciplinary Faculty, Safi Colloque Franco-Maghrébin d Analyse Stochastique Nice, France, Novembre 23-25, 25. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma / 4
2/4. Properties of fbm A Gaussian process B = {B t, t [, T ]} is called fractional Brownian motion(fbm) of urst parameter if it has zero mean and the covariance function R (t, s) := IE(B tb s) = 2 (t2 + s 2 t s 2 ), for s, t [, T ]. The fractional Brownian motion has the following properties: Self-similarity: for any constant a >, the processes {a B at, t } and {B t, t } has the same probability distribution. 2 Stationary increments: B t+s B s B t, s, t. From these properties, it follows that for every α > IE ( B t B s α ) = IE ( B α ) t s α. As a consequence of the Kolmogorov continuity theorem, we deduce that there exists a version of the fbm B which is a continuous process and whose paths are γ-ölder continuous for every γ <. The parameter controls the regularity of the trajectories of fbm. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 2 / 4
3/4. Stochastic integral with respect to fbm The fbm with urst parameter is not a semimartingale and then the Itô approach to the 2 construction of stochastic integrals with respect to fbm is not valid. Two main approaches have been used in the literature to define stochastic integrals with respect to fbm with urst parameter. Pathwise Riemann-Stieltjes stochastic integrals can be defined using Young s integral [6] in the case > 2. When ( 4, ), the rough path analysis introduced by Lyons [] is 2 a suitable method to construct pathwise stochastic integrals. 2 A second approach to develop a stochastic calculus with respect to the fbm is based on the techniques of Malliavin calculus. The divergence operator, which is the adjoint of the derivative operator, can be regarded as a stochastic integral, which coincides with the limit of Riemann sums constructed using the Wick product. This idea has been developed by Decreusefond and Üstünel [6], Carmona, Coutin and Montseny [4], Alòs, Mazet and Nualart [, 2], Alòs and Nualart, among others. The integral constructed by this method has zero mean. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 3 / 4
4/4. Malliavin calculus for fbm Let B = {B t, t [, T ]} be a fractional Brownian motion with urst parameter (, ) defined in a complete probability space (Ω, F, P), where F is generated by B. We denote by the ilbert space associated to B, defined as the closure of the linear space generated by the indicator functions { [,t], t [, T ]}, with respect to the inner product [,t], [,s] = R (t, s), s, t [, T ]. The mapping [,t] B t can be extended to a linear isometry between and the Gaussian T space generated by B. We denote by B(ϕ) = ϕtdbt the image of an element ϕ by this isometry. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 4 / 4
5/4. Malliavin calculus for fbm For a smooth and cylindrical random variable F = f (B(ϕ ),..., B(ϕ n)), with ϕ i and f Cb (IRn ) (f and all its partial derivatives are bounded), the derivative of F is the -valued random variable defined by DF = n j= f x j (B(ϕ ),..., B(ϕ n))ϕ j. For any integer k and any real number p we denote by D k,p the Sobolev space defined as the the closure of the space of smooth and cylindrical random variables with respect to the norm F p k,p = IE( F p ) + k j= IE( D j F p j ). Similarly, for a given ilbert space V we can define Sobolev spaces of V -valued random variables D k,p (W ). El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 5 / 4
6/4. Malliavin calculus for fbm The divergence operator δ is introduced as the adjoint of the derivative operator. More precisely, an element u L 2 (Ω; ) belongs to the domain of δ, denoted by Dom δ, if there exists a constant c u depending on u such that IE( DF, u ) c u F 2, for any smooth random variable F S. For any u Dom δ, δ(u) is the element of L 2 (Ω) given by the duality relationship IE(δ(u)F ) = IE( DF, u ), for any F D,2. We will make use of the property F δ(u) = δ(fu) + DF, u, () which holds if F D,2, u Dom δ and the right-hand side is square integrable. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 6 / 4
7/4. Malliavin calculus for fbm The covariance of the fractional Brownian motion can be written as t s R (t, s) = K (t, u)k (s, u)du, where K (t, s) is a square integrable kernel, defined for < s < t < T. In what follows, we assume that < <. In this case, this kernel has the following expression 2 [ ( ) t 2 K (t, s) = c (t s) 2 ( t ] s 2 )s 2 u 3 2 (u s) 2 du, s ( 2 with c = and β(x, y) := t x ( t) y dt for x, y >. Notice ) 2 ( 2)β( 2,+ 2 ) also that K t (t, s) = c ( ( ) t 2 ) 2 (t s) 2 3. s From these expressions it follows that the kernel K satisfies the following two estimates K t (t, s) c (t s) 3 2, (2) and K (t, s) d ((t s) 2 + s 2 ), (3) El assan for some Essakyconstant Poly-disciplinary d. Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 7 / 4
8/4. Malliavin calculus for fbm Let E be the linear span of the indicator functions on [, T ]. Consider the linear operator K from E to L 2 ([, T ]) defined by K (ϕ)(s) = K (T, s)ϕ(s) + T s (ϕ(t) ϕ(s)) K (t, s)dt. (4) t Notice that K ( [,t])(s) = K (t, s) [,t] (s). As a consequence C γ ([, T ]) L 2 ([, T ]). El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 8 / 4
9/4. Malliavin calculus for fbm It should be noted that the operator K is an isometry between the ilbert space and L 2 ([, T ]). That is, for every ϕ, ψ, ϕ, ψ = K ϕ, K ψ L 2 ([,T ]). (5) In this case the fbm is more irregular than the classical Brownian motion, and some ölder continuity is required for a function to be integrable. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 9 / 4
/4. Malliavin calculus for fbm Consider the following seminorm on the space E ϕ 2 K = T ϕ 2 (s)[(t s) 2 + s 2 ]ds T ( T 2 + ϕ(t) ϕ(s) (t s) 3 2 dt) ds. s (6) We denote by K the completion of E with respect to this seminorm. From the estimates (2) and (3), there exists a constant k such that for any ϕ K, ϕ 2 = K (ϕ) 2 L 2 ([,T ]) k ϕ 2 K. (7) As a consequence, the space K is continuously embedded in. This implies also that D,2 ( K ) D,2 () Dom δ. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma / 4
/4. Malliavin calculus for fbm One can show also that = I 2 T (L2 ([, T ])) L ([, T ]) (see Decreusefond -Üstünel [6]), where I 2 T ϕ(s) = Γ( 2 ) t (s t) 3 2 ϕ(s)ds is the left-sided fractional operator. For >, consider the Banach space of measurable functions ϕ : [, T ] IR 2 T T ϕ 2 = C r u 2 2 ϕ u ϕ r dudr <. In this case, we have L 2 ([, T ]) L ([, T ]). El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma / 4
2/4. q-variation of a stochastic process Fix q and T > and set ti n need the following definition. := it, where n is a positive integer and i =,, 2,..., n. We n Definition Let X be a given stochastic process defined in the complete probability space (Ω, F, P). Let V q n (X) be the random variable defined by n Vn q (X) := n i X q, where n i X := X t i+ n X t i n. We define the q-variation of X as the limit in L (Ω), as n goes to infinity, of Vn q (X) if this limit exists. i= El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 2 / 4
3/4. -variation of fbm Using the self-similarity of fbm and the Ergodic Theorem one can prove[ that the ] fbm has a finite -variation on any interval [, T ], equals to T e, where e = IE B (see, for instance, Rogers [5]). More precisely, we have, as n tends to infinity In fact, the self-similarity property implies that the sequence n B t n B i+ t n L (Ω) T e i. (8) i= n B t n B i+ t n i i= has the same distribution as T n n B i+ B i, i= and by the Ergodic Theorem this converges in L and almost surely to T e. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 3 / 4
4/4. Objective of the talk This result has been generalized by Guerra and Nualart [9] to the case of divergence integrals with respect to the fbm with urst parameter (, ). They have proved that 2 T V n (X) L (Ω) e u s ds, as n tends to infinity, where X t = t usδbs and e = IE [ ] B. The purpose of this talk is to study the -variation of divergence processes X = {Xt, t [, T ]} with respect to the fbm with urst parameter < 2. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 4 / 4
5/4. L p -estimates for the divergence integral with respect to fbm Let V be a given ilbert space. We introduce the following hypothesis for a V -valued stochastic process u = {u t, t [, T ]}, for some p 2. ypothesis (A.) p Let p 2. Then, sup s T u s L p (Ω;V ) < and there exist constants L >, α < 2 and γ > such that, 2 u t u s L p (Ω;V ) Ls α t s γ, for all < s t T. For any a < b T, we will make use of the notation u p,a,b = sup u s L p (Ω;V ). a s b The following lemma is a crucial ingredient to establish the L p -estimates for the divergence integral with respect to fbm. Lemma Let u = {u t, t T } be a process with values in a ilbert space V, satisfying assumption (A.) p for some p 2. Then, there exists a positive constant C depending on, γ and p such that for every < a b T IE ( u [a,b] p V ) C ( u p p,a,b (b a)p + L p a pα (b a) pγ+p). (9) El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 5 / 4
6/4. L p -estimates for the divergence integral with respect to fbm Idea of the proof. Suppose first that a >. By equalities (5) and (4) we obtain ( ) ( ) IE u [a,b] p V = IE K (u [a,b]) p L 2 ([,T ];V ) ( T ( ) = IE K K p (T, s)u s [a,b] (s) + u t [a,b] (t) u s [a,b] (s) t (t, s)dt s Consider the decomposition T ( ) [ b K u t [a,b] (t) u s [a,b] (s) t (t, s)dt = s s [ T ] [ b K + u s t (t, s)dt [a,b] (s) + Therefore b := I + I 2 + I 3. ( ) 3 IE u [a,b] p V C A i, a L 2 ([,T ];V ) (u t u s) K t (t, s)dt ] [a,b] (s) u t K t (t, s)dt ] [,a] (s) [ ] where A = IE K (T, )u [a,b] p and for i =, 2, 3, A L 2 ([,T ];V ) i = IE i= [ ] I i p. L 2 ([,T ];V ) ). El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 6 / 4
7/4. L p -estimates for the divergence integral with respect to fbm ypothesis (A.2) p Let u D,2 () be a real-valued stochastic process, which satisfies ypothesis (A.) p with constants L u, α and γ for a fixed p 2. We also assume that the -valued process {Du s, s [, T ]} satisfies ypothesis (A.) p with constants L Du, α 2 and γ for the same value of p. ypothesis (A.2) p means that u s and Du s have bounded L p norms in [, T ] and satisfy u t u s L p (Ω) L us α t s γ () Du t Du s L p (Ω;) L Du s α 2 t s γ, (2) for all < s t T. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 7 / 4
8/4. L p -estimates for the divergence integral with respect to fbm Theorem Suppose that u D,2 () is a stochastic process satisfying ypothesis (A.2) p for some p 2. Let < a b T. Then, there exists a positive constant C depending on, γ and p such that ( b p) IE u sδb s a C ( ( u p p,a,b + Du p p,a,b )(b a)p + (L p ua pα + L p Du a pα 2 )(b a) pγ+p). (3) If a =, then ( b p) ( IE u sδb s C ( u p p,a,b + Du p p,a,b )bp + (L p ub pα + L p Du b pα 2 )b pγ+p). (4) El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 8 / 4
9/4. L p -estimates for the divergence integral with respect to fbm Proof. We have ( b p) ( IE u sδb s C p IE( u[a,b] p ) + IE( Ds(ut [a,b](t)) ) p. a The first and the second terms of the above inequality can be estimated applying Lemma 2 to the processes u and Du, with V = R and V =, respectively. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 9 / 4
2/4. -variation for fbm ypothesis (A.3) Let u D,2 () be a real-valued stochastic process which is bounded in L q (Ω) for some q > and satisfies the ölder continuity property () with p =, that is u t u s L (Ω) L us α t s γ. (5) Suppose also that the -valued process {Du s, s [, T ]} is bounded in L (Ω; ) and satisfies the ölder continuity property (2) with p =, that is Du t Du s L (Ω;) L Du s α 2 t s γ. (6) Moreover, we assume that the derivative {D tu s, s, t [, T ]} satisfies sup D su t Kt α 3, (7) s T L (Ω) for every t (, T ] and for some constants < α 3 < 2 and K >. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 2 / 4
2/4. -variation for fbm Consider the indefinite divergence integral of u with respect to the fbm B, given by t X t = u sδb s := δ(u [,t] ). (8) Theorem Suppose that u D,2 () is a stochastic process satisfying ypothesis (A.3), and consider the divergence integral process X given by (8). Then, we have T V n (X) L (Ω) e u s ds, [ ] as n tends to infinity, where e = IE B. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 2 / 4
22/4. -variation for fbm We need to show that the expression n F n := IE i= t n i+ t n i u sδb s converges to zero as n tends to infinity. Using (), we can write T e u s ds, t n i+ u sδb s = t n i+ (u s u t n i )δb s + t n i+ u t n i δb s t n i = t i n t n i+ t n i t n i (u s u t n i )δb s Du t n i, [t n i,t n i+ ] + u t n i (B t n i+ B t n i ). := A,n i A 2,n i + A 3,n i. (9) By the triangular inequality, we obtain ( n F n IE A,n i A 2,n i + A 3,n i A 3,n i i= ) + D n, (2) El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 22 / 4
23/4. -variation for fbm where ( n ) T D n = IE A 3,n i e u s ds. i= Using the mean value theorem and ölder inequality, we can write ( n ) IE A,n i A 2,n i + A 3,n i A 3,n i i= ( n [ ] ) IE A,n i A 2,n i A,n i A 2,n i + A 3,n i + A 3,n i C [ IE i= ( n i= [ IE A,n i A 2,n i ( n i= )] A,n i A 2,n i + A 3,n i ) + IE ( n i= A 3,n i )]. (2) El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 23 / 4
24/4. -variation for fbm Substituting (2) into (2) yields F n CA n (Bn + Cn) + D n, where A n = IE B n = IE C n = IE ( n i= ( n i= ( n A,n i A 2,n i ) A,n i A 2,n i + A 3,n i A 3,n i i= ( n ) T D n = IE A 3,n i e u s ds. i= ), ), We first prove that B n and C n are bounded and second we show that A n converges to zero. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 24 / 4
25/4. -variation for fbm Remark that T B n = IE n =: K n + K n 2. u sδb s n + IE i= t n i+ t n i u sδb s Using estimate (4) with p =, it follows that ( K n C ( C u,, T n ) ( ) + Du n + L,, T u n α + L Du n α 2 n γ n n + n α γ + n α 2 ) γ. Therefore, K n is bounded since α < γ + and α 2 < γ +. Each term A 2,n i can be expressed as T ) A 2,n i = D su t n (R(s, t ni+ i s ) (R(s, tni ) ds. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 25 / 4
26/4. -variation for fbm In order to show that the term D n converges to zero as n tends to infinity, we replace n by the product nm and we let first m tend to infinity. That is, we consider the partition of interval [, T ] given by = t nm < < tnm = T and we define nm n Z n,m := u t nm nm i i B e u t n (t n j j+ tn j ) i= j= n [ (j+)m ( ) = u t nm u i t n nm j i B j= i=jm ( (j+)m ) ]. + u t n nm j i B e (tj+ n tn j ) i=jm n (j+)m u t nm u i t n j nm i B j= i=jm n (j+)m + u t n j nm i B e (tj+ n tn j ) := Z n,m + Z n,m 2. j= i=jm El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 26 / 4
27/4. -variation for fbm and for any n, Therefore, it follows from (22) and (23) that lim sup IE(Z n,m ) =, (22) n m lim IE(Z n,m 2 ) =. (23) m lim lim IE(Z n,m ) =. (24) n m El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 27 / 4
28/4. -variation for fbm Example Suppose that f : R R is a twice continuously differentiable function, such that f (x) + f (x) + f (x) e λx2, for some λ < 2T. Suppose that <. Then the process u = {f (Bt), t [, T ]} satisfies 2 4 Assumption (A.3) with constants α = α 2 = α 3 = and γ =. As a consequence, Theorem t 4. holds for the process X t = f (Bs)δBs. Indeed, let us just show condition (5). We can write, by the mean value theorem, ) f (B t) f (B s) L (Ω) (λ exp max t T B2 t B t B s, and we conclude using ölder s inequality. L (Ω) El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 28 / 4
29/4. -variation for d-fbm Consider a d-dimensional fractional Brownian motion (d 2) B = {B t, t [, T ]} = {(B () t, B (2) t,..., B (d) t ), t [, T ]} with urst parameter (, ) defined in a complete probability space (Ω, F, P), where F is generated by B. That is, the components B (i), i =,..., d, are independent fractional Brownian motions with urst parameter. We can define the derivative and divergence operators, D (i) and δ (i), with respect to each component B (i), as in Section 2. Denote by D,p i () the associated Sobolev spaces. We assume that these spaces include functionals depending on of all the components of B and not only the ith component. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 29 / 4
3/4. -variation for d-fbm Lemma Let F be a bounded random variable with values in IR d. Then, we have [ T ] V n ( F, B ) L (Ω) F, ξ ds ν(dξ), IR d as n tends to infinity, where ν is the normal distribution N(, I) on IR d. Theorem Suppose that for each i =,..., d, u (i) D,2 () is a stochastic process satisfying ypothesis (A.3). Set u t = (u () t,..., u (d) t ) and consider the divergence integral process d t X = {X t, t [, T ]} defined by X t := i= u(i) s δb s (i). Then, we have [ T ] V n (X) L (Ω) u s, ξ ds ν(dξ), IR d as n tends to infinity, where ν is the normal distribution N(, I) on IR d. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 3 / 4
3/4. Extended Domain The space is too small for some purposes. For instance, it has been proved in Cheridito-Nualart [5], that the trajectories of the fbm B belongs to if and only if > 4. This creates difficulties when defining the divergence δ(u) of a stochastic process whose trajectories do not belong to, for example, if u t = f (B t) and <, because the domain of δ 4 is included in L 2 (Ω; ). To overcome this difficulty, we extend the domain of the divergence operator. The main ingredient in the definition of this extended domain is the extension of the inner produce ϕ, ψ to the case where ψ E and ϕ L β ([, T ]) for some β > 2. More precisely, for ϕ L β m ([, T ]) and ψ = j= b j [,tj ] E we set ϕ, ψ = m T b j j= ϕ s R s (s, t j)ds. (25) This expression coincides with the inner produce in if ϕ, and it is well defined, because T ( ϕ, [,t] = R T ϕ s s (s, t)ds ϕ L β ([,T ]) sup t T ) R α s (s, t j) α ds <. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 3 / 4
32/4. Extended Domain We need the following extension of the domain of the divergence operator to processes with trajectories in L β ([, T ], IR d ), where β > 2. Definition Fix β >. We say that a d-dimensional stochastic process 2 u = (u (),..., u (d) ) L (Ω; L β ([, T ], IR d )) belongs to the extended domain of the divergence Dom δ, if there exists q > such that d IE u, DF d = IE( u (i), D (i) F ) cu F L q (Ω), (26) i= for every smooth and cylindrical random variable F S d, where c u is some constant depending on u. In this case δ(u) L p (Ω), where p is the conjugate of q, is defined by the duality relationship IE( u, DF ) = IE(δ(u)F ), for every smooth and cylindrical random variable F S d. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 32 / 4
33/4. Fractional Bessel process Let B be a d-dimensional fractional Brownian motion (d 2). The process R = {R t, t [, T ]}, defined by R t = B t, is called the fractional Bessel process of dimension d and urst parameter. It has been proved in [5] that, for >, the fractional Bessel 2 process R has the following representation R t = d i= t B s (i) t δb s (i) s 2 + (d ) ds. (27) R s R s This representation (27) is similar the one obtained for Bessel processes with respect to standard Brownian motion (see, for instance, Karatzas and Shreve [2]). Indeed, if W is a d-brownian motion and R t = W t, then R t = d i= t W s (i) dw s (i) + d t ds. R s 2 R s El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 33 / 4
34/4. Itô formula Theorem Let B a d-dimensional fractional Brownian motion with urst parameter <. Suppose that 2 F C 2 (IR d ) satisfies the growth condition { max F (x), F } (x), x IR d x i 2 F xi 2 (x), i =,..., d ce λx2, (28) where c and λ are positive constants such that λ < T 2. Then, for each i =,..., d and 4d F t [, T ], the process [,t] (B t) Dom E δ, and the following formula holds x i d t F F (B t) = F () + (B s)δb s (i) + x i= i d t 2 F x 2 (B s)s 2 ds, (29) i= i where Dom E δ is the extended domain of the divergence operator in the sense of Definition 3.9 in u, Jolis, Tindel [7]. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 34 / 4
35/4. Fractional Bessel process The next result is a change of variable formula for the fractional Bessel process in the case < 2. Theorem Let <, and let R = {Rt, [, T ]} be the fractional Bessel process. Set u(i) 2 t = B(i) t and R t u t = (u () t,..., u (d) t ), for t [, T ]. Then, we have the following results: (i) For any t (, T ], the process {u s [,t] (s), s [, T ]} belongs to the extended domain Dom δ and the representation (27) holds true. (ii) If > 4, for any t [, T ], the process u [,t] belongs to L 2 (Ω; d ) and to the domain of δ in L p (Ω) for any p < d. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 35 / 4
36/4. Fractional Bessel process Idea of the proof. i) Let us first prove part (i). Since the function x is not differentiable at the origin, the Itô formula (29) cannot be applied and we need to make a suitable approximation. For ε >, consider the function F ε(x) = ( x 2 + ε 2 ) 2, which is smooth and satisfies some growth condition. Applying Itô s formula we have F ε(b t) = ε + d i= t B (i) s (R 2 s + ε2 ) 2 t δb s (i) + d s 2 t s 2 Rs 2 ds ds. (3) (Rt 2 + ε2 ) 2 (Rs 2 + ε2 ) 3 2 El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 36 / 4
37/4. Fractional Bessel process Taking into account that δ(u ε [,t] ) converges to G t in L p, and that lim IE( u ε [,t], DF d ) = IE( u [,t], DF d ), ε since the components of u are bounded by one, we deduce that IE( u [,t], DF d ) = IE(G tf ). This implies that u [,t] belongs to the extended domain of the divergence and δ(u [,t] ) = G t. ii) Assume that > 4. We first show that for any i =,..., d, u(i) L 2 (Ω; ). We have IE( u (i) t 2 ) k IE +k IE ( T ( T ) (u s (i) ) 2 [(T s) 2 + s 2 ]ds ( T s 2 u (i) t u s (i) (t s) 3 2 dt) ds ). El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 37 / 4
38/4. Fractional Bessel process Theorem Suppose that 2d 2 >. Let R = {R t, t [, T ]} be the fractional Bessel process. Then, for i =, 2,..., d, the process u (i) t = B(i) t R t satisfies ypothesis (A.3). Idea of the proof : Fix i =,..., d. The random variable u (i) t is bounded and so, it is bounded in L q (Ω) for all q >. The Malliavin derivative D(i) u (i) is given by ( D s (i) u (i) t = Rt 3 (B (i) t ) ) 2 + Rt [,t] (s) := φ t [,t] (s). El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 38 / 4
39/4. Fractional Bessel process We now discuss the properties of the process Θ = {Θ t, t [, T ]} defined by Θ t := d i= t B s (i) δb s (i). R t We have that for every i =,..., d, u (i) t Therefore, we have the following corollary. Corollary Suppose that 2d 2 >. Then we have the following [ T V n (Θ) L (Ω) IR d = B(i) t satisfies ypothesis (A.3) if 2d 2 >. R t Bs, ξ R s ds ] ν(dξ), as n tends to infinity, where ν is the normal distribution N(, I) on IR d. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 39 / 4
4/4. Fractional Bessel process Proposition The process Θ is -self-similar. Let a >. By the representation (27) and the self-similarity of fbm, we have where the symbol that Θ is -self-similar. at s 2 Θ at = R at (d ) ds R O s t d = a R t (d )a u 2 du = a Θ t, R u d = means that the distributions of both processes are the same. This proves El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 4 / 4
4/4. Open problem : process of eigenvalues of a symmetric matrix-valued process Open problem : Consider now a family of independent fractional Brownian motion with urst parameter ) 4, 2 (, b = {(b ij(t), t ), i j d}. We define the symmetric matrix fbm B(t) with urst parameter by: B ij (t) = b ij if i < j and B ii (t) = 2b ii (t). We identify B(t) as an element of IR d(d+) 2. Lemma For every i =,..., d, there exists a function Φ i : IR d(d+) 2 IR, which is C in an open subset G of IR d(d+) 2, with G c =, such that λ i (t) = Φ i (b(t)). (Nualart-Abreu > 2 case). To study : u i t = Φ i x kh (b(t)), λ i (t) = λ i () + Y i t + 2 j i t s 2 λ i (s) λ j (s) ds. Y i is the divergence integral of u i. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 4 / 4
4/4 Alòs, E., Mazet, O., Nualart, D., Stochastic calculus with respect to fractional Brownian motion with urst parameter lesser than 2. Stoch. Proc. Appl. 86 (999), 2 39. Alòs, E., Mazet, O., Nualart, D., Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2), 766 8. Alòs, A., Nualart, D., Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (22), 29 52. Carmona, P., Coutin, L., Montseny, G., Stochastic integration with respect to fractional Brownian motion. Ann. Inst. enri Poincaré 39 (23), 27 68. Cheridito, P., Nualart, D., Stochastic integral of divergence type with respect to fractional Brownian motion with urst parameter in (, /2). Ann. Inst. enri Poincaré 4 (25), 49 8. Decreusefond, L., Üstünel, A. S., Stochastic analysis of the fractional Brownian motion. Potential Anal. (998), 77 24. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 4 / 4
4/4 u, Y., Jolis, M., Tindel, S., On Stratonovich and Skorohod stochastic calculus for Gaussian processes. Ann. Probab. 4 (23), no. 3A, 656 693. urst,. E., Long-term storage capacity in reservoirs. Trans. Amer. Soc. Civil Eng. 6 (95), 4 4. Guerra, J., Nualart, D., The /-variation of the divergence integral with respect to the fractional Brownian motion for > 2 and fractional Bessel processes. Stochastic Process. Appl. 5 (25), 9-5. Kolmogorov, A. N., Wienersche Spiralen und einige andere interessante Kurven im ilbertschen Raum. (German) C. R. (Doklady) Acad. URSS (N.S.) 26 (94), 5 8. Lyons, T., Differential equations driven by rough signals. Rev. Mat. Iberoamericana 4 (998), 25 3. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 4 / 4
4/4 Karatzas, I., Shreve, S. E., Brownian Motion and Stochastic Calculus. New York, Springer-Verlag (99). Mandelbrot, B. B., Van Ness, J. W., Fractional Brownian motions, fractional noises and applications. SIAM Review (968), 422 437. Nualart, D., The Malliavin calculus and related topics. Springer Verlag. Second Edition (26). Rogers, L. C. G., Arbitrage with fractional Brownian motion. Math. Finance 7 (997), no., 95 5. Young, L. C., An inequality of the ölder type connected with Stieltjes integration. Acta Math. 67 (936), 25 282. El assan Essaky Poly-disciplinary Faculty (Cadi the Ayyad -variation University of the divergence Poly-disciplinary integral Faculty, Safi Colloque Franco-Ma 4 / 4