Discrete approximation of stochastic integrals with respect to fractional Brownian motion of Hurst index H > 1 2

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1 Discrete approximation of stochastic integrals with respect to fractional Brownian motion of urst index > 1 2 Francesca Biagini 1), Massimo Campanino 2), Serena Fuschini 2) 11th March 28 1) 2) Department of Mathematics, LMU, Theresienstr. 39 D-8333 Munich, Germany fax number: biagini@math.lmu.it Department of Mathematics, University of Bologna, Piazza di Porta S. Donato, 5 I-4127 Bologna, Italy fax number: campanin@dm.unibo.it, fuschini@dm.unibo.it Abstract In this paper we provide a discrete approximation for the stochastic integral with respect to the fractional Brownian motion of urst index > 1 defined in terms of the divergence operator. To 2 determine the suitable class of integrands for which the approximation holds, we also investigate the relations among the spaces of Malliavin differentiable processes in the fractional and standard case. AMS 2 subject classifications: Primary 6G15, 6G18, 6XX, 67, 64. Key words and phrases: Fractional Brownian motion, Malliavin derivative, divergence, stochastic integral. 1 Introduction Fractional Brownian motion (fbm) of urst index (, 1) has been widely used in applications to model a number of phenomena, e.g. in biology, meteorology, physics and finance. A natural question is how to define a stochastic integral with respect to fbm and how to provide a natural discretization of it. The problem of defining a stochastic integral with respect to the fractional Brownian motion has been extensively studied in literature. Several different integrals has been introduced by exploiting different properties of the fractional Brownian motion: as Wiener integrals (exploiting the Gaussianity) in [12], [13], [24], as pathwise integrals (using fractional calculus and forward, backward and symmetric integrals) in [2], [3], [8], [9], [11], [2], [23], [26], [27], [28], as divergence (using fractional Malliavin calculus) in [3], [4] [1], [24], as stochastic integral with respect to the standard Brownian motion in [6], [16], as stochastic integral with respect to the fractional white noise in [14], [17]. For a complete survey on this subject, an extensive literature and relations among different definitions of integrals we refer to [7] and [24]. For applications, in particular for applications to finance, it is relevant to give an interpretation to a definition of stochastic integral. As it is the case of Itô and Skorohod integrals, this can be achieved by giving a procedure to obtain the integral as the limit of discrete approximations that are meaningful in the application field. 1

2 In this paper we address the problem of finding a discrete approximation of the stochastic integral defined as divergence. A discrete realization of the fractional Brownian motion for urst index > 1 2 is provided in different ways in [13] and [19]. This can be extended to find a discretization of stochastic integral of deterministic functions integrable with respect to the fractional Brownian motion. In [5] they study the convergence of Riemann-Stieltjes integrals with respect to the fractional Brownian motion for urst index > 1 2 when the integrands are almost surely ölder-continuous of order λ > 1. Other discrete approximations of stochastic integrals with respect to the fractional Brownian motion can be found in [3], [8], [14] and [22]. ere we focus on the case > 1 2 and consider the stochastic integral with respect to B defined in terms of the divergence operator δ relative to the fbm. We prove a discrete approximation for this kind of integral by means of the resolutions of the Fock space associated to B and by using its expression in terms of the divergence operator δ of the standard Brownian motion. In order to find a suitable class of stochastic integrands for which our results hold, we also study the relations among the spaces of Malliavin differentiable processes with respect to B and to B with urst index > Basic definitions and notation We recall in this section the basic definition and the main properties concerning fractional Brownian motion and the relative Malliavin calculus. Definition 2.1. The fractional Brownian motion (fbm) B of urst index (, 1) is a centered Gaussian process {B t } t with covariance given by R (s, t) := 1 2 (s2 + t 2 t s 2 ). (1) As well-known, for = 1 2 we obtain the standard Brownian motion (Bm). By Kolmogorov s theorem we have that B admits a continuous modification for every (, 1). In the sequel we assume to work always with the continuous modification of B, that we denote again with B. In particular we will focus on the case > 1 2. In this case the covariance (1) can be written as where φ is given by R (s, t) = α t s φ(r, u)drdu (2) φ(r, u) = r u 2 2 (3) and α = (2 1). By [24], we obtain that R (s, t) can be expressed in terms of the deterministic kernel t K (t, s) = c s 1 2 (u s) 3 2 u 1 2 du (4) where c = ţ ű 1 (2 1) 2 β(2 2,, t > s, in the following way 1 2 ) R (t, s) = t s s K (t, u)k (s, u)du. As in the standard Brownian motion case, we wish to associate to B a Gaussian ilbert space in the following way. We denote E the space of step functions defined on [, T ] and let be the ilbert space given by the completion of E with respect to the following inner product χ [,t], χ [,s] = R (t, s). 2

3 The function χ [,t] Bt can be extended to an isometry between and the Gaussian ilbert space generated by the fractional Brownian motion. We denote this isometry by B. We recall that the following relation holds since L 2 ([, T ]) (5) f k f L 2 ([,T ]) (6) for every f L 2 ([, T ]), where k = T For the proof we refer to [4] and [24]. 2 We introduce now the operator K. Definition 2.2. Let K such that We note that be the linear operator defined on E (K ϕ)(s) := Proposition 2.3. For every ϕ, ψ E we have Proof. For the proof we refer to [4] and [23]. K : E L 2 ([, T ]) T s K (χ [,t] )(s) = K (t, s)χ [,t] (s). ϕ(r) K (r, s)dr. (7) r K ϕ, K ψ L 2 ([,T ]) = ϕ, ψ. (8) ence we can extend the operator K to an isometry between the ilbert space and L2 ([, T ]) (see [24] for further details). We define the process B t = B ((K ) 1 χ [,t] ) t [, T ]. (9) We obtain that B t is a Gaussian process, whose covariance is given by B t, B s L 2 ([,T ]) = B ((K ) 1 χ [,t] ), B ((K ) 1 χ [,s] ) L 2 ([,T ]) = (K ) 1 (χ [,t] ), (K ) 1 (χ [,s] ) = s t ence B t (or better its continuous modification) is a standard Brownian motion. Moreover for every ϕ we have and in particular B (ϕ) = T B t = B (χ [,t] ) = (K ϕ)(t)db t T K (t, s)db s, (1) where the kernel K is defined in (4). As a consequence of (9) and (1), we immediately have that B and B generate the same filtration. We denote by S the set of regular cylindric random variables S of the form S = f(w (h 1 ),..., W (h n )), (11) where h i, n 1, and f belongs to the set Cb (Rn, R) of bounded continuous functions with bounded continuous partial derivatives of every order. 3

4 Definition 2.4. Let [ 1 2, 1) and S S. (1) The Malliavin derivative of S with respect to B is defined as the valued random variable D S := n i=1 (2) The space S is endowed with the norm D 1,2 S 2 D 1,2 f x i (W (h 1 ),..., W (h n ))h i. (12) such that := S 2 L 2 (Ω) + D S 2 L 2 (Ω,). We denote by D 1,2 the closure of the space S with respect to the norm D 1,2 and extend by density the derivative operator to the space D 1,2. We introduce now the divergence operator. Let [ 1 2, 1). Definition 2.5. (1) The domain domδ of the divergence operator is defined as the set of the random variables F L 2 (Ω, ) such that for some constant c F depending on F. E( D S, F ) c F S L 2 (Ω) S S, (13) (2) The divergence operator δ : domδ L 2 (Ω) is defined by the duality relation for every G D 1,2. D G, F L 2 (Ω,) = G, δ F L 2 (Ω) (14) ence the divergence operator is the adjoint of the derivative operator. From now on, to simplify the notation in the case = 1 2, we omit to write the index and denote L 1,2 := D 1,2 1/2 (L2 ([, T ])). We conclude this section by recalling a result that will play a key role in the sequel. Proposition 2.6. We have that F domδ iff K F domδ and Proof. For the proof we refer to [3] and [24]. δ (F ) = δ(k F ). (15) According to the approach of [12] and [15], we define the stochastic integral with respect to B for > 1 2 as follows. Definition 2.7. Let B be a fractional Brownian motion of urst index > 1 2 and F a stochastic process belonging to domδ. We define stochastic integral of F with respect to B the divergence δ (F ) of F introduced in Definition Relations between the standard case and the case > 1 2 Before proceeding to prove our main results concerning the discretization of the stochastic integral as divergence, we need to clarify some relations between the standard and the fractional case. 4

5 3.1 Chaos expansions By the Chaos Expansion Theorem, for every F L 2 (Ω, L 2 ([, T ])) we have t [, T ], F t = n I q (f q,t ) in L 2 (Ω) (16) where f q,t (t 1,..., t q ) := f q (t 1,..., t q, t) L 2 ([, T ] q+1 ) are symmetric with respect to the first q variables (t 1,..., t q ) and I q denotes the q-dimensional iterated Wiener integral with respect to (t 1,..., t q ) I q (f q,t ) := T ( tq... t2 f q,t (t 1,..., t q )db t1... db tq 1 ) db tq (for further details see [18] and [23]). We introduce the function K : L 2 (Ω, ) L 2 (Ω, L 2 ([, T ])) (17) that associates to F L 2 (Ω, ) the random variable K F L2 (Ω, L 2 ([, T ])) defined as (K F )(ω) := (F (ω)). K Proposition 3.1. The following properties hold (i) K : L2 (Ω, ) L 2 (Ω, L 2 ([, T ])) is an isometry; (ii) L 2 (Ω, L 2 ([, T ])) L 2 (Ω, ) and Proof. The proof follows by (5), (6) and (8). F L2 (Ω,) k F L2 (Ω,L 2 ([,T ])). Proposition 3.2. Let F belong to L 2 (Ω, L 2 ([, T ])) with chaos expansion given by (16). Then K F = Moreover for almost every t [, T ] where by (7) we have I q (K, (f q, )) in L 2 (Ω, L 2 ([, T ])). (18) (K F ) t = (K, f q, ) t = I q ((K, f q, ) t ) in L 2 (Ω), (19) T Proof. We start with some preliminary remarks. (A) If we put t (S Q ) t := f q (t 1,..., t q, r) K (r, t)dr. r Q I q (f q,t ), Q N, then by the Chaos Expansion Theorem (see [23]) we obtain the following convergences lim (S Q) t F t L Q 2 (Ω) = t [, T ] (2) lim S Q F L Q 2 (Ω,L 2 ([,T ])) =. (21) 5

6 (B) Consider F = K F. Then F admits chaos expansion given by F = I q (fq, ) in L 2 (Ω, L 2 ([, T ])), where fq, L 2 ([, T ] q+1 ) are symmetric function with respect to the variables t 1,..., t q, for every q N. ence if we set Q (SQ) t := I q (fq,t), Q N, the following convergences hold lim Q (S Q) t Ft L 2 (Ω) = (22) lim Q S Q F L 2 (Ω,L 2 ([,T ])) =. (23) (C) By the stochastic version of the Fubini-Tonelli theorem ([25]), we can exchange the order of integration between the multiple integral I q and the integral operator K for every t [, T ]. By (8) and Proposition 3.1 we obtain that [K, (I q ((f q, ))] t = I q (K, (f q, ))(t), (24) ence by (21) we have Now it remains only to prove that By (23) and (25) we get immediately K S Q K F 2 L 2 (Ω,L 2 ([,T ])) = K (S Q F ) 2 L 2 (Ω,L 2 ([,T ])) k S Q F 2 L 2 (Ω,L 2 ([,T ])). lim Q K S Q K F L 2 (Ω,L 2 ([,T ])) =. (25) lim Q (K S Q ) t (K F ) t L 2 (Ω) =, for a.e. t [, T ]. (26) lim Q S Q K S Q 2 L 2 (Ω,L 2 ([,T ])) =. (27) Moreover we know Q SQ K S Q 2 L 2 (Ω,L 2 ([,T ])) = I q (fq, Kf q, ) and that the following relation holds between the norms: 2 L 2 (Ω,L 2 ([,T ])). (28) I q (f q,t ) 2 L 2 (Ω,L 2 ([,T ])) = q! (f q,t) 2 L 2 ([,T ] q+1 ), (29) 6

7 if (f q,t ) is symmetric with respect to the first q variables t 1,..., t q, for every q N. Since also f q, K, fq, are symmetric with respect to (t 1,..., t q ), for every q N, we get and by (27) Q I q (fq, K, f q, ) 2 L 2 (Ω,L 2 ([,T ])) = Q q! fq, K, f q, 2 L 2 ([,T ] q+1 ) q! fq, K, f q, 2 L 2 ([,T ] q+1 ) =. Finally ( T fq, K, f q, 2 L 2 ([,T ] q+1 ) = for every q N. We conclude that for almost every t [, T ] [,T ] q (f q, K, f q, ) 2 (t 1,..., t q, t)dt 1... dt q f q,t (K, f q, ) t = in L 2 ([, T ] q ) ) dt = for every q N. Finally for almost every t [, T ] we have that (SQ ) t = (K S Q) t, for every Q [, T ], and then lim Q (K S Q ) t (K F ) t L 2 (Ω) = lim Q (S Q) t (K F ) t L 2 (Ω). By (22) we obtain the result. 3.2 Relation between D 1,2 () and L1,2 We can generalize the Definition 2.5 to the case of valued random variables. In this case the space S () is defined as the set of random variables U with values in of the form U = n F j v j, (3) j=1 where F j D 1,2, v j, j {1,..., n}. If U S (), the Malliavin derivative D U of U is defined as the element in L 2 (Ω, ) given by D U := n D F j v j, (31) j=1 where denotes the ilbert-tensor product between the ilbert spaces. We refer to [1] for the definition and the properties of the tensor product. The derivative operator D : S () L 2 (Ω, ) induces on S () the norm We extend Definition 2.5 as follows. U 2 D 1,2 () := U 2 L 2 (Ω,) + D U 2 L 2 (Ω, ). 7

8 Definition 3.3. The space D 1,2 () is defined as the closure of S () with respect to the norm D 1,2 The derivative operator D can be extended to D 1,2 (). We note that the space D 1,2 () is an ilbert space with respect to the inner product for every F, G D 1,2 (). In particular we have Proposition 3.4. F, G D 1,2 () := F, G L 2 (Ω,) + D F, D G L 2 (Ω, ), (32) D 1,2 Proof. For the proof and further details, see [18]. () domδ and δ F L 2 (Ω) F D 1,2 (). Consider now the operator (K ) 2 : L 2 (Ω, ) L 2 (Ω, L 2 ([, T ] 2 ), defined on F L 2 (Ω, ) as follows (K ) 2 (ω) := (K ) 2 (F (ω)). (). Proposition 3.5. The operator K (i) Let F L 2 (Ω, ). Then and has the following properties: F D 1,2 () K F L 1,2 (33) (K ) 2 (D F ) = D(K F ). (34) (ii) K : D1,2 () L1,2 is an isometry with respect to the inner product defined in (32), i.e. K F, K G L 1,2 = F, G D 1,2 () for every F, G D 1,2 (). Proof. (i) Let F D 1,2 () L2 (Ω, ). Then exists a sequence (U n ) in S (), such that { U n F in L 2 (Ω, ) D U n D F in L 2 (Ω, ) (35) for n. We recall that by (3) the term U n is given by U n = k n j=1 F n j h n j n N where F n j D1,2 and h j, j {1,..., n}. We define U n := K U n, n N. By Proposition 3.1 and (35) we have lim U n F L n 2 (Ω,) =. That implies for n. U n K F in L 2 (Ω, L 2 ([, T ])) (36) 8

9 In [24] is proved that D 1,2 = D1,2 and in particular By (37) we obtain the following relation K (D F ) = DF, F D 1,2 = D1,2. (37) (K ) 2 (D U n ) = k n DFj n j=1 K h n j = D(U n). Then D(U n) (K ) 2 (D F ) 2 L 2 (Ω,L 2 ([,T ] 2 ) = = (K ) 2 (D U n ) (K ) 2 (D F ) 2 L 2 (Ω,L 2 ([,T ] 2 )) = D U n D F 2 L 2 (Ω, ). The last term goes to zero by (35) and we have D(U n) (K ) 2 (D F ) in L 2 (Ω, L 2 ([, T ] 2 )) (38) for n. By (36) and (38) we obtain that (34) holds. If we now assume K F L1,2, we can proceed in the same way using (K ) 1 instead of K. ence we have proved also (33). (ii) This result follows immediately by (32), and from that the fact that K is an isometry. We can prove now the following result, that is essential to characterize the space of integrands F, whose stochastic integral δ (F ) admits the discretization provided in Section 4. Proposition 3.6. The following inclusion holds and where k = T as in (6). L 1,2 D 1,2 () (39) F D 1,2 () k F L 1,2 F L 1,2. (4) Proof. Let F L 1,2. We recall that it is equivalent to the condition q(q!) f q, ) 2 L 2 ([,T ] q+1 ) < (41) (see [23]). By Proposition 3.2 and Proposition 3.5 we have that F belongs to D 1,2 () iff q(q!) K, f q, ) 2 L 2 ([,T ] q+1 ) <. By (6) we have K, f q, 2 L 2 ([,T ] q+1 ) = [,T ] q f q (t 1,..., t q ) 2 dt 1...dt q k f q 2 L 2 ([,T ] q+1 ) 9

10 and then q(q!) K, f q, ) 2 L 2 ([,T ] q+1 ) k q(q!) f q 2 L 2 ([,T ] q+1 ). (42) Since F L 1,2, the series q(q!) f q 2 L 2 ([,T ] q+1 ) is convergent in L2 (Ω, L 2 ([, T ])). Moreover (42) implies that converges in L 2 (Ω, L 2 ([, T ])). We prove now (4). We note that and ence by (42), it follows that Finally we get q(q!) K, f q, ) 2 L 2 ([,T ] q+1 ) DF 2 L 2 (Ω,L 2 ([,T ] 2 )) = q(q!) f q 2 L 2 ([,T ] q+1 ) D(K F ) 2 L 2 (Ω,L 2 ([,T ] 2 )) = q(q!) K, f q, 2 L 2 ([,T ] q+1 ). D(K F ) 2 L 2 (Ω,L 2 ([,T ] 2 )) k DF 2 L 2 (Ω,L 2 ([,T ] 2 )) F 2 D 1,2 () = F 2 L 2 (Ω,) + D F 2 L 2 (Ω, ) = K F 2 L 2 (Ω,L 2 ([,T ])) + (K ) 2 (D F ) 2 L 2 (Ω,L 2 ([,T ] 2 )) = K F 2 L 2 (Ω,L 2 ([,T ])) + D(K F ) 2 L 2 (Ω,L 2 ([,T ] 2 )) k F 2 L 2 (Ω,L 2 ([,T ])) + k DF 2 L 2 (Ω,L 2 ([,T ] 2 )) = k F 2 L 1,2. Remark 3.7. The space L 1,2 is strictly included in D 1,2 () L2 (Ω, L 2 ([, T ])). We prove this by showing that there exists an element F D 1,2 () L2 (Ω, L 2 ([, T ])) that doesn t belong to L 1,2. Let F have the following chaos expansion F = I q (f q, ) in L 2 (Ω, L 2 ([, T ])) where Suppose that f satisfies the following hypotheses: (i) f ; f q,t (t 1,..., t q ) = 1 1 T q q!q β f(qα t) α, β >. (43) (ii) f L 2 ([, )); (iii) f(u)f(v)φ(u, v)dudv <, where φ is defined in (3). 1

11 Since by (18) and (33) the following must hold F L 1,2 q(q!) f q, 2 L 2 ([,T ] q+1 <, F D 1,2 () q(q!) K, f q, 2 L 2 ([,T ] q+1 <, it is sufficient to prove that there exist f q, of the form (43) such that By standard integral calculations we get the following inequalities and q(q!) f q, 2 L 2 ([,T ] q+1 ) =, (44) q(q!) K, f q, 2 L 2 ([,T ] q+1 ) <. (45) where c f = f(u)f(v)φ(u, v)dudv. Finally (44) and (45) hold if we choose α e β such that { α + β 2, 2α + β > 2. Since > 1 2, it is then sufficient to choose α = β = 1. q(q!) f q, 2 L 2 ([,T ] q+1 ) 1 f 2 L 2 ([,T ]) q α+β 1 (46) q(q!) f q, 2 L 2 ([,T ] q+1 ) c 1 f. (47) q2α+β 1 4 Discrete approximation for the stochastic integral We can now prove a discrete approximation for the stochastic integral defined as divergence introduced in Definition 2.7. By Proposition 2.6 it follows that the integral of F with respect to fbm coincides the Skorohod integral of K F with respect to the Bm. ence one can provide a discretization of the integral δ (F ) by using a discrete approximation of δ(k F ). owever this is not satisfactory because the discretization depends on the urst index of the Brownian motion with respect to which the integral is defined. Therefore we define first a discretization of the process and then a procedure for obtaining the approximation of the integral. 4.1 Resolution of [, T ] We introduce first the setting and some basic notions needed to prove the main result of this paper. For further details we refer to [21]. Consider the measure space ([, T ], B, l), where B denotes the Borelian σ-algebra and l the Lebesgue measure. (48) 11

12 Definition 4.1. A resolution of L 2 ([, T ]) is an increasing family of σ-algebras generated by a finite number of sets B 1 B 2 B n B. such that L 2 ([, T ], B n, l) is dense in L 2 ([, T ]). We denote with Γ n the finite set of generators of B n, n N that we assume to be a partition of the interval [, T ]. Given a resolution {B n }, consider the set Γ N := N n= Γ n and the set Γ = n N Γ n. We obtain on R Γ a coherent system of marginal laws θ N and by Kolmogorov theorem there exists a probability measure θ on R Γ with this family of marginal laws. ence we can associate a probability space (R Γ, θ) to a resolution {B n }. Proposition 4.2. The probability spaces associated to different resolutions are canonically isomorphic. Proof. For the proof see [21]. 4.2 Fock space We denote with R the set of all resolutions of L 2 ([, T ]). Definition 4.3. The Fock space is the set of all pairs (Ω(r), Ψ r,r ) where r, r R, Ω(r) denotes the probability space associated to the resolution r e Ψ r,r is the canonical isomorphism between Ω(r) and Ω(r ). All properties of a Fock space do not depend on the choice of the resolution. For further details, see [21]. Consider a subset γ B. The space L 2 ([, T ]) can be decomposed as follows L 2 ([, T ]) = L 2 (γ) L 2 (γ c ). This decomposition induces a similar one on the relative Fock spaces Definition 4.4. We denote by Foc(L 2 ([, T ])) = Foc(L 2 (γ)) Foc(L 2 (γ c )). E γc : L 2 (Foc(L 2 ([, T ]))) L 2 (Foc(L 2 (γ c ))) (49) the natural projection from L 2 (Foc(L 2 ([, T ]))) onto L 2 (Foc(L 2 (γ c ))). We recall the construction of the Skorohod-Zakai-Nualart-Pardoux integral (in short, SZNP-integral) given in [21] for F L 2 (Ω, L 2 ([, T ])). We define I n (F ) := (E Bn (F ))(γ)b(γ), E γ c γ Γ n where {B n } is a resolution of L 2 ([, T ]) as introduced in Definition 4.1, E γc denotes the projection (49) and B(γ) is the usual Wiener integral of the indicator function χ γ with respect to Bm, i.e. B(γ) := The SZNP-integral is defined as the limit By [21] we have the following theorem T χ γ (s)db s, γ B. I(F ) := lim n I n(f ) in L 2 (Ω). 12

13 Theorem 4.5. Let F L 1,2. Then the SZNP-integral exists and Proof. For the proof we refer to [21]. lim I n(f ) = δ(f ) in L 2 (Ω). (5) n Let F L 1,2. Since by Proposition 3.6, L 1,2 D 1,2 () and by Proposition 3.4, D1,2 () domδ, the stochastic integral δ (F ) is well defined. Definition 4.6. Let F L 1,2. We define In (F ) := E γ c (E Bn (K F )(γ))b(γ), γ Γ n Jn (F ) := E B n (E γ c (K (EB n (F )))(γ)b(γ), γ Γ n where {B n } is a resolution of L 2 ([, T ]) as introduced in Definition 4.1 and E γc denotes the projection (49). By Theorem 4.5 we have that and by Proposition 2.6 we get lim n I n (F ) = δ(k F ) in L 2 (Ω), (51) δ(k F ) = δ (F ). The purpose is now to prove that J n (F ) provides the desired discretization of δ (F ). Theorem 4.7. Let F L 1,2. Then lim J n (F ) = δ (F ) in L 2 (Ω). n Proof. Let F L 1,2. We need to prove ( I n (F ) Jn (F ) ) = in L 2 (Ω). lim n By (16) the chaos expansion of F is given by F t = I q (f q,t ), where f q,t L 2 ([, T ] q + 1) are symmetric functions for every q N. By Proposition 3.2 we get (K F ) t = I q (K, f q, )(t). We have that E Bn (K F ) t = E B n (K F )(γ) = E γc E B n (K F )(γ) = I q (E Bn (K, f q, )(t), (52) I q (E B n (f q, )(γ)), (53) I q ( E B n (K, f q, )(γ)χ q γ c ). (54) 13

14 ere we write K, fq, to indicate on which variable the operator K is operating. By [23] we obtain that the product of two Wiener integrals can be written as ence I n (F ) can be rewritten as and analogously I q ( E B n (K, f q, ))(γ)χ q γ c ) B(γ) = Iq+1 ( E B n (K, f q, ))(γ)χ γ χ q γ c ). I n (F ) = J n (F ) = ( I q+1 ( I q+1 γ c ) γ Γ n E Bn (K, f q, ))χ γ χ q γ Γ n E Bn ( K, (E B n (f q, )) ) χ γ χ q γ c ). It follows that the difference between I n (F ) and J n (F ) is of the form I n (F ) J n (F ) = ( I q+1 γ Γ n Taking the norms, we get In (F ) Jn (F ) 2 L 2 (Ω) = (q +1)! γ Γ n Let A n : [, T ] q+1 R be the function [ E B n (K, f q, ) E B n ( K, (E B n (f q, )) )] χ γ χ q γ c ). [ E B n (K, f q, )) E Bn ( K, (E Bn (f q, )) )] χ γ χ q γ c A n := E B n (K, f q, E B n ( K, (E B n (f q, ) ). 2 L 2 ([,T ] q+1 ). Since for every n N, the sets {γ} γ Γn 2 γ c γ Γ n A n χ γ χ q L 2 ([,T ] q+1 ) are a partition of the interval [, T ], we obtain ( ) 2dt1 = A n χ γ χ q γ... dt c q dt [,T ] q+1 γ Γ n ( = A 2 nχ γ χ q γ )dt c 1... dt q dt [,T ] q+1 γ Γ n ( ) A 2 nχ γ dt 1... dt q dt [,T ] q+1 γ Γ n ( ) = A n (t 1,... t q, t) χ γ (t) dt 1... dt q dt [,T ] q+1 γ Γ n = A n 2 L 2 ([,T ] q+1 ). ence it follows In (F ) Jn (F ) 2 L 2 (Ω) (q + 1)! A n 2 L 2 ([,T ] q+1 ). Put A q (n) := (q + 1)! A n 2 L 2 ([,T ] q+1 ). To prove that A q (n) goes to zero in L 2 (Ω) for n, it is sufficient to verify the following conditions: A q (n) as n q, (55) 14

15 Condition (55) is equivalent to We compute the norm A q (n) converges uniformly in n. (56) A n 2 L 2 ([,T ] q+1 ) = To prove (57), we define the sequence lim A n 2 n L 2 ([,T ] q+1 ) =. (57) = A 2 n(t 1,..., t q, t)dt 1... dt q dt [,T ] q+1 ( T ) A 2 n(t 1,..., t q, t)dt dt 1... dt q. [,T ] q G n (t 1,..., t q ) := T A 2 n(t 1,..., t q, t)dt (58) and show that it verifies the hypothesis of Lebesgue Theorem on bounded convergence. Indeed G n (t 1,..., t q ) = A n 2 L 2 ([,T ]) = = E Bn (K, f q, )) E Bn( K, (E Bn (f q, )) ) 2 L 2 ([,T ]) = E B n [ (K, f q, )) (K, (E B n (f q, ))) ] 2 L 2 ([,T ]) K, f q, K, (E B n (f q, )) 2 L 2 ([,T ]) = K, ( fq, E Bn (f ) q, ) 2 L 2 ([,T ]) = f q, E Bn (f q, )) 2 k f q, E Bn (f q, )) 2 L 2 ([,T ]). where k = T as in (6). The last term tends to zero when n and then lim G n(t 1,..., t q ) =, for a.e. (t 1,..., t q ). n We need now to show that there exists a function G L 1 ([, T ] q ) such that The following relation holds G n G, for a.e. (t 1,..., t q ). G n (t 1,..., t q ) = G n (t 1,..., t q ) c f q, E B n (f q, )) 2 L 2 ([,T ]) c ( fq, L 2 ([,T ]) + E Bn (f q, )) L 2 ([,T ]) 2 4c f q, 2 L 2 ([,T ]). 15

16 Set G(t 1,..., t q ) := 4c f q 2 L 2 ([,T ]). We have [,T ] q G(t 1,..., t q )dt 1... dt q = 4c = 4c [,T ] q ( [,T ] q f q 2 L 2 ([,T ]) dt 1... dt q T f 2 q (t 1,..., t q, t)dt)dt 1... dt q = 4c f q 2 L 2 ([,T ] q+1 ) <, (59) since F L 1,2. ence the norm G L 1 ([,T ] q ) is finite and by the Lebesgue theorem applied to G n (t 1,..., t q ) we can conclude that A n 2 L 2 ([,T ] q+1 ) converges to zero when n. This ends the proof of (57) and hence of (55). To prove (56) we use the following criterion of uniform convergence if sup A q (n) <, n then A q (n) converges uniformly in n. We have by (58) and (59) that ence A q (n) = (q + 1)! A n 2 L 2 ([,T ] q+1 ) = (q + 1)!G n (t 1,..., t q ) (q + 1)!4c f q 2 L 2 ([,T ] q+1 ). sup A q (n) 4c (q + 1)! f q 2 L 2 ([,T ] q+1 ). n Since F L 1,2, the series f q 2 L 2 ([,T ] q+1 ) is convergent and then also sup n A q (n) <. This ends the proof. Acknowledgments We wish to express our gratitude to Paul Malliavin for interesting discussions and remarks. References [1] J.P. Aubin, Applied functional analysis, John Wiley and Sons, [2] E. Alòs, A. León, D. Nualart, Stratonovich stochastic calculus with respect to to fractional Brownian motion with urst parameter less than 1/2, Taiwanese Journal of Mathematics 5, , 21. [3] E. Alòs, O. Mazet, D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29, , 2. [4] E. Alòs, D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastics Reports 75, , 23. [5] T. O. Androshchuk, Y. S. Mishura, The sequence of semimartingales that converges to the process of capital in a mixed Brownian-fractional Brownian model, Preprint University of Kiev,

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