Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions

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1 Sochasic inegraion wih respec o mulifracional Brownian moion via angen fracional Brownian moions Eric Herbin, Joachim Lebovis, Jacques Lévy Véhel To cie his version: Eric Herbin, Joachim Lebovis, Jacques Lévy Véhel. Sochasic inegraion wih respec o mulifracional Brownian moion via angen fracional Brownian moions <hal v5> HAL Id: hal hps://hal.inria.fr/hal-65388v5 Submied on 26 Nov 212 (v5), las revised 15 Jan 213 (v6) HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Sochasic inegraion wih respec o mulifracional Brownian moion via angen fracional Brownian moions Eric Herbin Joachim Lebovis Jacques Lévy Véhel November 18, 212 Absrac Sochasic inegraion w.r.. fracional Brownian moion (fbm) has raised srong ineres in recen years, moivaed in paricular by applicaions in finance and Inerne raffic modelling. Since fbm is no a semi-maringale, sochasic inegraion requires specific developmens. Mulifracional Brownian moion (mbm) generalizes fbm by leing he local Hölder exponen vary in ime. This is useful in various areas, including financial modelling and biomedicine. The aim of his wor is wofold: firs, we prove ha an mbm may be approximaed in law by a sequence of angen" fbms. Second, using his approximaion, we show how o consruc sochasic inegrals w.r.. mbm by ransporing" corresponding inegrals w.r.. fbm. We illusrae our mehod on examples such as he Wic-Iô, Sorohod and pahwise inegrals. Keywords: Fracional and mulifracional Brownian moions, Gaussian processes, convergence in law, whie noise heory, Wic-Iô inegral, Sorohod inegral, pahwise inegral. 1 Moivaion and Bacground Fracional Brownian moion (fbm) is a cenred Gaussian process wih feaures ha mae i a useful model in various applicaions such as financial and Inerne raffic modeling, image analysis and synhesis, physics, geophysics and more. These feaures include self-similariy, long range dependence and he abiliy o mach any prescribed consan local regulariy. Is covariance funcion R H reads: R H (, s) := γ H 2 ( 2H s 2H s 2H ), where γ H is a posiive consan and H, which is usually called he Hurs exponen, belongs o (, 1). When H = 1 2, fbm reduces o sandard Brownian moion. Various inegral represenaions of fbm are nown, including he harmonizable and moving average ones [25], as well as represenaions by inegrals over a finie domain [2, 9]. The fac ha mos of he properies of fbm are governed by he single real H resrics is applicaion in some siuaions. In paricular, is Hölder exponen remains he same all along is rajecory. This does no seem o be adaped o describe adequaely naural errains, for insance. In addiion, long range dependence requires H > 1/2, and hus imposes pahs smooher han he ones of Brownian moion. Mulifracional Brownian moion was inroduced o overcome hese limiaions. The basic idea is o replace he real H by a funcion h() ranging in (, 1). Several definiions of mulifracional Brownian moion exis. The firs ones were proposed in [21] and in [4]. A more general approach was inroduced in [26]. In his wor, we shall use a new definiion ha includes all previously nown ones and which is, in our opinion, boh more flexible and reains he essence of his class of Gaussian processes. We firs need o define a fracional Brownian field: Regulariy eam, INRIA Saclay and MAS Laboraory, Ecole Cenrale Paris, Grande Voie des Vignes, Chaenay- Malabry Cedex, France. eric.herbin@ecp.fr, joachim.lebovis@ecp.fr and jacques.levy-vehel@inria.fr. Laboraoire de Probabiliés e Modèles Aléaoires, C.N.R.S (UMR 7599), Universié Pierre e Marie Curie (Paris VI), case 188, 4, pl. Jussieu, F Paris Cedex 5, France. 1

3 Definiion 1.1 (Fracional Brownian field). Le (Ω, F, P ) be a probabiliy space. A fracional Brownian field on R (, 1) is a Gaussian field, noed (B(, H)) (,H) R (,1), such ha, for every H in (, 1), he process (B H ) R defined by B H := B(, H) is a fracional Brownian moion wih Hurs parameer H 1. A mulifracional Brownian moion is simply a pah raced on a fracional Brownian field. precisely, i is defined as follows: More Definiion 1.2 (Mulifracional Brownian moion). Le h : R (, 1) be a deerminisic coninuous funcion. A mulifracional Brownian moion (mbm) wih funcional parameer h is he Gaussian process B h := (B h ) R defined by B h := B(, h()) for all in R. A word on noaion: B Ḥ or B. h() will always denoe an fbm wih Hurs index H or h(), while B ḥ will sand for an mbm. Noe ha B h := B(, h()) = B h(), for every real. The funcion h is called he regulariy funcion of mbm. I is sraighforward o chec ha any mulifracional Brownian moion in he sense of [26, Def.1.1] is also an mbm wih our definiion. Fracional fields (B(, H)) (,H) R (,1) leading o previously considered mbms include: B 1 (, H) := 1 c H B 3 (, H) := R R e iu 1 W Ä u H1/2 1 (du), B 2 (, H) := u H 1/2 u H 1/2ä W 2 (du), R ä T W 3 (du), B 4 (, H) := 1 { u< T } (, u) K H (, u) W 4 (du), Ä H 1/2 ( u) ( u) H 1/2 where c H := ( 2 cos(πh)γ(2 2H) )1/2 H(1 2H), dh := ( 2HΓ(3/2 H) 1/2 Γ(1/2H)Γ(2 2H)) and K H (, s) := d H ( s) H 1/2 c H (1/2 H) s (u s) H 3/2( 1 ( s u) 1/2 H ) du, and where, for i {1; 2; 3; 4}, W i denoes an independenly scaered sandard Gaussian measure on R, and W 1 denoes he complex-valued Gaussian measure which can be associaed in a unique way o W 1 (see [26, p.23-24] and [25, p ] for more deails). Replacing H wih h() in B 1 (, H) and B 2 (, H) leads o he so-called harmonisable mbm, firs considered in [4]. The same operaion on B 3 (, H) yields he moving average mbm defined in [21]. Boh are paricular cases of mbms in he sense of [26]. Finally, B 4 (, h()) corresponds o he Volerra mulifracional Gaussian process sudied in [9]. This las process is an mbm in our sense. The definiion of a fracional Brownian field does no specify is iner-line behaviour, i.e. he relaions beween (B H ) R and (B H ) R for H H. In order o obain a useful heory, we need o conrol hese relaions o some exen. I urns ou ha he following condiion is sufficien o prove all he resuls we will need in his paper. We shall denoe E[Y ] he expecaion of a random variable Y in L 1 (Ω, F, P ). (H 1 ) : [a, b] R, [c, d] (, 1), (Λ, δ) (R ) 2, such ha E[(B(, H) B(, H )) 2 ] Λ H H δ, for all (, H, H ) in [a, b] [c, d] 2. Using he equaliy E[(B(, H) B(s, H)) 2 ] = s 2H and he riangular inequaliy for he L 2 -norm, Assumpion (H 1 ) is seen o be equivalen o he following one: (H) : [a, b] [c, d] R (, 1), (Λ, δ) (R ) 2, s.. E[(B(, H) B(s, H )) 2 ] Λ Ä s 2c H H δä, for all (, s, H, H ) [a, b] 2 [c, d] 2. Thus, we will refer eiher o assumpion (H 1 ) or (H) in he sequel. 1 Alernaively, one migh sar from a family of fbms (B H ) H (,1) (i.e. B H := (B H) R is an fbm for every H in (, 1)) and define from i he field (B(, H)) (,H) R (,1) by B(, H) := B H. However i is no rue, in general, ha he field (B(, H)) (,H) R (,1) obained in his way is Gaussian. 2

4 Remar 1. (i) Assumpion (H) enails ha he map (, s, H, H ) E[B(, H) B(s, H )] is coninuous on R 2 (, 1) 2. (ii) I is well-nown ha, since B is Gaussian, Assumpion (H) and Kolmogorov s crierion enail ha he field B has a d-hölder coninuous version for any d in (, δ 2 c). In he sequel we will always wor wih such a version. In many cases, B will be specified hrough an inegral represenaion. I is hus relevan o recas assumpion (H) in erms of he ernel used in hese represenaions. We disinguish beween wo siuaions: he case where he inegral is over a compac inerval, and where i is over R. Inegral on a compac se [, T ] In his siuaion (see, e.g [2]), he fracional field (B(, H)) (,H) [,T ] (,1) is defined by B(, H) := T K(, u, H) W(du), where W is a Gaussian measure, K is defined on [, T ] 2 (, 1) and is such ha u K(, u, H) belongs o L 2 ([, T ], du), for all (, H) in [, T ] (, 1). This is for insance he case of B 4. As one can easily see, he following condiion (C K ) enails (H): (C K ) : (c, d) wih < c < d < 1, H K(, u, H) is Hölder coninuous on [c, d], uniformly in (, u) in [, T ] 2, i.e (M, δ) (R ) 2, (, u) in [, T ] 2, K(, u, H) K(, u, H ) M H H δ. Condiion (C K ) is fulfilled by he ernel K defining B 4 (see [9, Proposiion 3, (5)]). Inegral over R A represenaion wih an inegral over R is used for insance in [7, 12]. The fracional field (B(, H)) (,H) R (,1) is hen defined by B(, H) := R M(, u, H) W(du) where M is defined on R 2 (, 1), and is such ha u M(, u, H) belongs o L 2 (R, du), for every (, H) in R (, 1). This is he case for he fields B 1, B 2 and B 3. Condiion (C M ) enails (H): (C M ) : [a, b] R, [c, d] (, 1), δ R, [a, b], Φ L 2 (R, du), verifying sup [a,b] R s.. (u, H, H ) R [c, d] 2, M(, u, H) M(, u, H ) Φ (u) H H δ. Condiion (C M ) is fulfilled by he ernel M defining B 1, B 2 and B 3. See Appendix A for a proof. Ouline of he paper Φ (u) 2 du <, The remaining of his paper is organized as follows. Our main resul in Secion 2 (Theorem 2.1 Poin 2 (i)) is ha an mbm may be approximaed in law (as well as in he L 2 and almos sure senses) by a sequence of angen fbms. In Secion 3 we show how o define a sochasic inegral w.r.. mbm as a limi of inegrals w.r.. approximaing fbms. The main resul is Theorem 3.3 ha provides a condiion on he sochasic inegral w.r.. fbm ha guaranees convergence of he sequence of approximaions. In oher words, as soon as a mehod of inegraion w.r.. fbm verifies his condiion, hen our mehod allows o ranspor" i ino an inegral w.r.. mbm. We apply his consrucion o he cases of he Wic-Iô, Sorohod and pahwise inegrals respecively in Secions 4, 5 and 6. 2 Approximaion of mulifracional Brownian moion Since an mbm is jus a coninuous pah raced on a fracional Brownian field, a naural quesion is o enquire wheher i may be approximaed by paching adequaely chosen fbms, and in which sense. 3

5 Heurisically, for a < b, we divide [a, b) ino small inervals [ i, i1 ), and replace on each of hese B h by he fbm B Hi where H i = h( i ). I seems reasonable o expec ha he resuling process i BHi 1 [i, i1)() will converge, in a sense o be made precise, o B h when he sizes of he inervals [ i, i1 ) go o. Our aim in his secion is o mae his line of hough rigorous. 2.1 Approximaion of mbm by piecewise fbms In he sequel, we fix a fracional Brownian field B and a coninuous funcion h, hus an mbm, noed B h. We aim o prove ha his mbm can be approximaed on every compac inerval [a, b] by paching ogeher fracional Brownian moions defined on a sequence of pariions of [a, b]. In ha view, we choose an increasing sequence (q n ) n N of inegers such ha q := 1 and 2 n q n 2 2n for all n in N. For a compac inerval [a, b] of R and n in N, le := { ; [[, q n]]} where := a (b a) q n for in [[, q n ]] (for inegers p and q wih p < q, [[p, q]] denoes he se {p; p 1; ; q}). Define, for n in N, he pariion A n := {[, x(n) 1 ); [[, q n 1]]} { q n }. I is clear ha A := (A n ) n N is a decreasing nesed sequence of subdivisions of [a, b] (i.e. A n1 A n, for every n in N). For in [a, b] and n in N here exiss a unique ineger p in [[, q n 1]] such ha p < p1. We will noe he real p in he sequel. The sequence ( ) n N is increasing and converges o as n ends o. Besides, define for n in N, he funcion h n : [a, b] (, 1) by seing h n (b) = h(b) and, for any in [a, b), h n () := h( ). The sequence of sep funcions (h n ) n N converges poinwise o h on [a, b]. Define, for in [a, b] and n in N, he process q n 1 B hn := B(, h n ()) = = 1 (n) [x )() B(, h(x(n),x(n) )) 1 {b}() B(b, h(b)). (2.1) 1 Noe ha, despie he noaion, he process B hn is no an mbm, as h n is no coninuous. We believe however here is no ris of confusion in using his noaion. B hn is almos surely càdlàg and disconinuous a imes, in [[, q n]]. The following heorem shows ha mbm appears naurally as a limi objec of sums of fbms: Theorem 2.1 (Approximaion heorem). Le B be a fracional Brownian field, h : R (, 1) be a coninuous deerminisic funcion and B h be he associaed mbm. Le [a, b] be a compac inerval of R, A be a sequence of pariions as defined above, and consider he sequence of processes defined in (2.1). Then: 1. If B is such ha he map C : (, s, H, H ) E[B(, H) B(s, H )] is coninuous on [a, b] 2 h([a, b]) 2 hen he sequence of processes (B hn ) n N converges in L 2 (Ω) o B h,i.e. [a, b], [Ä lim E ä B h 2 ] n B h =. 2. If B saisfies assumpion (H) and if h is β-hölder coninuous for some posiive real β, hen he sequence of processes (B hn ) n N converges (i) in law o B h, i.e. (ii) almos surely o B h, i.e. {B hn ; [a, b]} P Å { [a, b], law {Bh ; [a, b]}. ã lim Bhn = B h } = 1. Before we proceed o he proof, we noe ha Poin 2 (i) above is a much sronger saemen han he well-nown localisabiliy of mbm, i.e. he fac ha he moving average (see [21]), harmonisable (see [4]) and Volerra mbms (see [9]) are all angens" o fbms in he following sense: for every real u, { B h } ur Bh u law ; [a, b] r h(u) r {Bh(u) ; [a, b]}. 4

6 Proof: The ouline of he proof is as follows: he only delicae poin is 2 (i). The difficuly lies in he fac ha he sequence of processes (B hn ) n N is no coninuous bu merely càdlàg. As he limi process is coninuous, his siuaion may be deal wih he heorem on page 92 in [22]. In order o show ighness, we shall mae use of [1, (2.6) p.43]. This requires finding an upper bound for he quaniy E [ sup B( i, H) B( i, h( i )) ], where K (i) denoes a compac se of [a, b]. This is done in Lemma 2.2, H K (i) using chaining argumens from [27]. Le us now presen he deailed proof. 1. Le [a, b]. For any n in N, one compues E î ( B h n B h ) 2 ó = C(,, h(x (n) ), h( )) 2 C(,, h( ), h()) C(,, h(), h()). The coninuiy of he maps h, (, H, H ) C(,, H, H ) and he fac ha lim x(n) = enail ha lim E î (B hn B h ) 2ó =. 2. By assumpion, here exiss (η, β) in R R such ha for all (s, ) in [a, b], h(s) h() η s β. (2.2) (i) We proceed as usual in wo seps (see for examples [8, 23]), a): finie-dimensional convergence and b): ighness of he sequence of probabiliy measures (P B hn ) n N. a) Finie dimensional convergence Since he processes B h and B hn defined by (2.1) are cenred and Gaussian, i is sufficien o prove ha lim E î ó [ ] B hn Bs hn = E B h Bs h for every (s, ) in [a, b] 2. The cases where = b or s = are consequences of poin 1. above. We now assume ha a s < < b. One compues E î ó B hn Bs hn = 1 )() [, )(s) E [B(, h n())b(s, h n (s))]. 1 j j1 Hence, E î B hn s s < B hn s enail ha lim E î B hn 1 (n) [x (,j) [,q n 1] 2,x(n) ó î (n) = E B(, h(x ))B(s, h( s )) ó for all large enough inegers n (i.e. such ha ). The coninuiy of h, (i) of Remar 1, and he fac ha lim ( B hn s ó [ ] = E B h Bs h., x s (n) ) = (, s), b) Tighness of he sequence of probabiliy measures (P B hn ) n N. We are in he paricular case where a sequence of càdlàg processes converges o a coninuous one. The heorem on page 92 of [22] applies o his siuaion: i is sufficien o show ha, for every posiive reals ε and τ, here exis an ineger m and a grid { i } i [,m] such ha a = < 1 < < m = b ha verify Ç å lim sup P max i m sup B hn [ i, i1) B hn i > τ < ε. (2.3) Le us hen fix (ε, τ) in (R ) 2. Define [H 1, H 2 ] := [ inf h(u), sup h(u)] and se, unil he end of his u [a,b] u [a,b] proof, q n := 2 2n, n N and F := [a, b] [H 1, H 2 ]. The process (B(, H)) (,H) F is Gaussian and he space of coninuous real-valued funcions defined on F endowed wih he sup-norm is a separable Banach space. Fernique s heorem ï (see [11, Theorem 2.6 p.37]) applies o he effec ha here exiss a posiive real α such ha A α := E exp { α sup B(, H) 2}ò <. Se G := L p= 2p/2 q δ/2 p, where L is he (,H) F universal consan given in [27, Theorem p. 33]. Define also D := Λ (η (b a) β ) δ where Λ and δ are he consans appearing in (H) and η in (2.2). Le N be he smalles ineger n such ha { max A α (1 q n ) exp α τ 2 4 b a 2H 2 q2h2 n ; 4 (1 qn ) exp τ D qδβ n ; b a q n ; G D1/2 q δβ/2 n } < 1 τ 8 ε 2. (2.4) 5

7 Se m := m(τ, ε) = q N and i := x (N) i for i in [[, m]]. Noe ha (2.4) enails ha, for all n larger han N, h n ( i ) = h( i ). Besides, also as soon as n N, h n () belongs o he se h([ i, i1 ]) when [ i, i1 ]. Le J τ,m n J τ,m n where and := P ({ max i m (1m) max P ({ i m sup [ i, i1) sup [ i, i1) B hn B hn i > τ }). Then B(, h n ()) B( i, h n ( i )) > τ }) (1m) ( ) max i m Lτ,i n max i m Qτ,i n, L τ,i n := P ({ sup B(, h n ()) B( i, h n ()) > τ/2 }) [ i, i1) Q τ,i n := P ({ sup B( i, h n ()) B( i, h n ( i )) > τ/2 } ). [ i, i1) Upper bound for (1 m) max i m Lτ,i n : The couple (i, n) being fixed in [[, m]] N, he process (B(s, h n ())) s [i, i1] is a fracional Brownian moion of Hurs index h n (). Using incremen-saionariy and self-similariy of fracional Brownian moion yields: L τ,i n = P ( sup Å P [ i, i1) sup v [,1] B( i, h n ()) > τ/2 ) = P ( B(v, h n ( i v( i1 i ))) > sup u [, i1 i) τ 2 i1 i H 2 Using Marov ideniy and Fernique s heorem we ge, L τ,i n P ( sup B(v, H) > (v,h) F E [ exp { α We have shown ha sup (,H) F Å ) τ 2 i1 i H 2 = P exp { α B(, H) 2}] exp { ατ 2 4 i1 i 2H 2 i1 i hn(ui) B ( (2.5) u i1 i, h n (u i ) ) ) > τ/2 ã. (2.6) sup B(v, H) 2 } > exp { } ã α τ 2 4 i1 i 2H 2 (v,h) F } = Aα exp { ατ 2 q 2H 2 } N 4 b a 2H 2 < ε 2(1q N ) = ε 2(1m). i [[, q N ]], n N, (1 m) max i m Lτ,i n < ε 2. (2.7) Upper bound for (1 m) max i m Qτ,i n : Fix a couple (i, n) in [[, m]] N. Recall ha h n () belongs o h([ i, i1 ]) =: K (i) for every in [ i, i1 ). We hence have, Q τ,i n P ({ sup H K (i) B( i, H) B( i, h( i )) > τ/2 }) 2 P ({ sup H K (i) =:X i(h) {}}{ B( i, H) B( i, h( i )) > τ/4 }). Observe ha he righ hand side of he previous inequaliy does no depend on n any more. Our aim is o apply [1, (2.6) p.43]. In ha view, we firs prove he following esimae: ï ò Lemma 2.2. For all i in [[, q N ]], µ i := E sup X i (H) < G D1/2 < τ H K (i) q δβ/2 8. N Proof of Lemma 2.2: Fix i in [[, q N ]]. We recall some noions of chaining from [27]. A sequence C (i) := (C n (i) ) n N of pariions of K (i) is called admissible if i is increasing and such ha card(c n (i) ) 2 2n, for every n in N. Le d i denoe he pseudo-disance associaed o he Gaussian process (X i (H)) H K (i), (i.e. d i (H, H ) := (E[(X i (H) X i (H )) 2 ]) 1/2, for (H, H ) in K (i) K (i) ). (2.8) 6

8 For (H, p) in K (i) N, le C (i) p diameer of C (i) p ha: where γ 2 (K (i), d i ) := inf (H) is by definiion i (C (i) sup H K (i) (H) be he unique elemen of he pariion C p (i) p (H)) := sup p which conains H. The d i (H, H ). [27, Theorem 2.1.1] enails (H,H ) C p(h) 2 µ i L γ 2 (K (i), d i ), (2.9) 2 p/2 i (C p (i) (H)) and where he infimum is aen over all admissible sequences of pariions of K (i). Le H (i) 1 and H (i) 2 be such ha K (i) =: [H (i) of pariions C (i) := (C (i) n 1, H(i) 2 ) n N of K (i) defined, for every ineger n, by seing C (i) n ]. Consider he sequence := {[y (n), y(n) 1 ); [[, q n 1]]} {y q (n) n }, where y (n) = H (i) 1 and y (n) 1 y(n) = H(i) 2 H(i) 1 q n, for (n, ) in N [[, q n 1]]. I is clear ha C (i) is a decreasing nesed sequence of pariions of K (i) and is hence admissible. For (H, p) in K (i) N, denoe [y (p), y (p) 1 ) he unique elemen of C(i) p which conains H. Then, using (H), µ i L p L p L Λ 1/2 p 2 p/2 sup sup H K (i) (H,H ) [y (p),y (p) 2 p/2 sup sup [,q p 1] (H,H ) [y (p),y(p) 2 p/2 sup y (p) [,q p 1] (E[(X i (H) X i (H )) 2 1/2 ]) 1 )2 1 y(p) (E[(X i (H) X i (H )) 2 1/2 ]) 1 )2 δ/2 = G Λ 1/2 (H (i) 2 H (i) 1 )δ/2. By Hölder coninuiy of h, H (i) 2 H (i) 1 η i1 i β = η b a β. Using (2.4), we finally ge µ i < GD 1/2 q δβ/2 N < τ/8. Noe ha Lemma 2.2 implies ha (τ/8) 2 < (τ/4 µ i ) 2. Le us now go bac o he proof of ighness. [1, (2.6) p.43] yields, for all n N, q β N Q τ,i n 4 exp { (τ/4 µ i) } 2 2σ, i 2 where σi 2 := sup E [ X i (H) 2]. By definiion of (X i (H)) H K (i) and assumpion (H): H K (i) σi 2 Λ sup H h( i ) δ Λ η δ i1 i βδ = Λ η δ βδ b a = D. H K (i) q βδ q βδ N N This yields ha Q τ,i n 4 exp { τ 2 q δβ N 2 7 D } and finally: i [[, q N ]], n N, (1 m) max i m Qτ,i n < ε 2. Using (2.7) and he previous inequaliy, Inequaliy (2.5) hen becomes Jn τ,m ends (i). (ii) Almos sure convergence < ε, for every n N, which Denoe Ω he measurable subse of Ω, verifying P ( Ω) = 1, such ha for every ω in Ω, (, H) B(, H)(ω) is coninuous on [a, b] [H 1, H 2 ]. Then, for every ω in Ω, we ge: B hn (ω ) B h (ω ) = B(, h n ())(ω ) B(, h())(ω ) = ( (n)) B(, h x )(ω ) B(, h())(ω ). This ends he proof. Remar 2. Wih some addiional wor, one may esablish he almos sure convergence of (B hn ) n N under he sole condiion of coninuiy of h. 7

9 3 Sochasic inegrals w.r.. mbm as limis of inegrals w.r.. fbm The resuls of he previous secion, especially 2 (i) of Theorem 2.1, sugges ha one may define sochasic inegrals wih respec o mbm as limis of inegrals wih respec o approximaing fbms. We formalize his inuiion in he presen secion. We consider as above a fracional field (B(, H)) (,H) R (,1), bu assume in addiion ha he field is C 1 in H on (, 1) in he L 2 (Ω) sense, i.e. we assume ha he map H B(, H), from (, 1) o L 2 (Ω), is C 1 for every real. We will denoe B (, H ) he L 2 (Ω)-derivaive a poin H of he map H B(, H). The field ( B(,H) ) (,H) R (,1) is of course Gaussian. We will need ha he derivaive field saisfies he same assumpion (H 1 ) as B(, H). More precisely, from now on, we assume ha B(, H) saisfies (H 2 ): (H 2 ) : For all [a, b] [c, d] R (, 1), H B(, H) is C 1 in he L 2 (Ω) sense from (, 1) o L 2 (Ω) for every in [a, b], and here exiss (, α, λ) (R ) 3 such ha, for all (, s, H, H ) in [a, b] 2 [c, d] 2, E î ( B B (, H) (s, H ) ) 2 ó Ä s α H H λä. Proposiion 3.1. The fracional Brownian fields B i := (B i (, H)) (,H) R (,1), i [[1, 4]], verify Assumpion (H 2 ). Proof: The proof of his proposiion in he case of B 1 and B 2 may be found in Appendix B. The ones for B 3 and B 4 are easily obained using resuls from [21] and [9] and are lef o he reader. In he remaining of his paper, we consider a C 1 deerminisic funcion h : R (, 1), a fracional field B which fulfills assumpions (H 1 ) and (H 2 ), and he associaed mbm B h := B(, h()). We now explain in a heurisic way how o define an inegral wih respec o mbm using approximaing fbms. Wrie he differenial of B(, H): db(, H) = B B (, H) d (, H) dh. Of course, his is only formal as B(, H) is no differeniable in he L 2 -sense nor almos surely wih respec o. I is, however, in he sense of Hida disribuions, bu we are no ineresed in his fac a his sage. Wih a differeniable funcion h in place of H, his (again formally) yields db(, h()) = B (, h()) d h () B (, h()) d. (3.1) The second erm on he righ-hand side of (3.1) is defined for almos every ω and every real by assumpion. Moreover, i is almos surely coninuous as a funcion of and hus Riemann inegrable on compac inervals. On he oher hand, he firs erm of (3.1) has no meaning a priori since mbm is no differeniable wih respec o. However, since sochasic inegrals wih respec o fbm do exis, we are able o give a sense o B (, H) for every fixed H in (, 1). Coninuing wih our heurisic reasoning, we hen approximae B (, h()) by lim qn 1 B )() (, h n()). This formally yields: db(, h()) q n 1 lim = = 1 (n) [x,x(n) 1 B 1 (n) [x,x(n) )() 1 (, h n()) d h () B (, h()) d. (3.2) Assuming we may exchange inegrals and limis, we would hus lie o define, for suiable processes Y, Y db(, h()) = q n 1 lim = x (n) 1 Y db h(x(n) ) Y h () B (, h()) d, (3.3) 8

10 where he firs erm of he righ-hand side of (3.3) is a limi, in a sense o be made precise depending on he mehod of inegraion, of a sum of inegrals wih respec o fbms and he second erm is a Riemann inegral or an inegral in a weaer sense (see Secion 4). In order o mae he above ideas more precise, le us fix some noaions. (M) will denoe a given mehod of inegraion wih respec o fbm (e.g Sorohod, whie noise, pahwise, ). For he sae of noaional simpliciy, we will consider inegrals over he inerval [, 1]. For H in (, 1), denoe Y d (M) B H he inegral of Y := (Y ) [,1] on [, 1] wih respec o he fbm B H, in he sense of mehod (M), assuming i exiss. The following noaion will be useful: Noaion (inegral wih respec o lumped fbms) Le Y := (Y ) [,1] be a real-valued process on [, 1] which is inegrable wih respec o all fbms of index H in h([, 1]) in he sense of mehod (M). We denoe he inegral wih respec o lumped fbms in he sense of mehod (M) by: Y d (M) B hn := q n 1 = 1 (n) [x,x(n) )() Y d (M) B h( ) 1, n N (3.4) (we use he same noaions as in subsecion 2.1: (q n ) n N is a sequence of inegers and he family := { ; [[, q n]]} is defined by := q n for in [[, q n ]]). Wih his noaion, our enaive definiion of an inegral w.r.. o mbm (3.3) reads: Y db(, h()) = lim Y d (M) B hn Y h () B (, h()) d, (3.5) The ineres of (3.3) is ha i allows o use any of he numerous definiions of sochasic inegrals wih respec o fbm, and auomaically obain a corresponding inegral wih respec o mbm. I is worhwhile o noe ha, wih his approach, an inegral wih respec o mbm is a sum of wo erms: he firs one seems o depend only on he chosen mehod for inegraing wih respec o fbm (for insance, a whie noise or pahwise Riemann inegral), while he second is an inegral which appears o depend only on he field used o define he chosen mbm, i.e. essenially on is correlaion srucure. This second erm will imply ha he inegral wih respec o he moving average mbm, for insance, is differen from he one wih respec o he harmonisable mbm. As he example of simple processes in he nex subsecion will show, he second erm does however also depend on he inegraion mehod wih respec o fbm. Noe ha he naure of Y d (M) B hn depends on (M). For example, Y d (M) B H and hence Y d (M) B hn will belong o L 2 (Ω) if (M) denoes he Sorohod inegral, whereas Y d (M) B H and hence Y d (M) B hn belong o he space (S) of sochasic disribuions when (M) denoes he inegral in he sense of whie noise heory. We will wrie Y d (M) B h for he inegral of Y on [, 1] wih respec o mbm in he sense of (M) (which is ye o be defined). When we do no wan o specify a paricular mehod bu insead wish o refer o all mehods a he same ime, we will wrie Y db hn and Y db h insead of Y d (M) B hn and Y d (M) B h. In order o gain a beer undersanding of our approach, we explore in he following subsecion he paricular cases of simple deerminisic and hen random inegrands. 3.1 Example: simple inegrands Deerminisic simple inegrands Any reasonable definiion of an inegral mus be linear. Thus, o deermine he inegral of deerminisic simple funcions w.r.. mbm, i suffices o consider he case of Y = 1. Obviously, we should find ha 1 1 dbh = B1 h. In order o verify his fac, le us compue he limi of he sequence ( 1 dbhn ) n N. Proposiion 3.2. Assume ha, almos surely, for all real, he real-valued map H B(, H)(ω) belongs o C 2 ((, 1)). The sequence ( 1 dbhn ) n N hen converges almos surely and in L 2 (Ω) o B(1, h(1)) (, h()) d, where he second erm is a pahwise inegral. h () B 9

11 Proposiion 3.2 implies ha, for regular enough fields B and h funcions, Formula (3.3) does indeed yield 1 dbh = B h 1. Proof: By definiion of 1 dbhn, for almos all ω, I n := 1 dbhn (ω) = B h(x(n) qn 1 ) 1 (ω) q n 1 =1 Denoe K n (ω) := q n 1 =1 ( h(x B (n) (ω) B h(x(n) ) 1 ) ( h(x B (n) (ω) B h(x(n) ) 1 ) (ω) ) B h() (ω). (3.6) (ω) ) Since boh H B(, H)(ω) and h are smooh, he finie incremens heorem applied wice allows one o wrie, for some θ and ϕ in ( 1, x(n) ), K n (ω) = q n 1 =1 q 1 n h (ϕ ) B (x(n), h(θ ))(ω). Denoe F := [, 1] h([, 1]). Using again he finie incremens heorem, he fac ha he processes ( B (, H)) (,H) F and ( 2 B (, H)) 2 (,H) F are Gaussian, and he uniform coninuiy of h we ge lim K n (ω) = h () B (, h()) d(ω). Equaliy (3.6) enails ha he sequence ( 1 dbhn ) n N converges almos surely o B(1, h(1)) h () B (, h()) d. I remains o prove ha he convergence also holds in L 2 (Ω). Firs, we remar ha 1 dbhn is in L 2 (Ω) since B is a Gaussian process. Le us now show ha h () B (, h()) d belongs o L2 (Ω). Cauchy-Schwarz inequaliy and Assumpion (H 2 ) enail: E [( h () B (, h()) d) 2] h 2 ( L 2 (R) ( α h() λ ) d ) <. Now, almos sure convergence of he sequence ( 1 dbhn ) n N implies convergence in L 2 (Ω) provided i is bounded by a random variable X L 2 (Ω). Assumpion (H) enails ha he sequence (B h(x(n) qn 1 ) 1 ) n N converges o B h(1) 1 in L 2 (Ω). I hus remains o sudy he random variable K n defined above. By almos sure coninuiy (which follows from Assumpion (H 2 )), he cenred Gaussian process ( B (, H)) (,H) F has bounded sample pahs wih probabiliy one. Moreover i is well-nown (see [1, (2.4) p.43] for example) ha his enails ha B (, H) belongs o L 2 (Ω). L 2 (Ω) convergence follows. sup (,H) F Remar 3. Proposiion 3.2 applies o he four fields considered in he inroducion Simple processes We now consider a paricular case of a simple process ha will show ha Formula (3.3) does no always yield he expeced resul, and mus be modified in cerain siuaions. We ae Y = Y wih Y a cenred Gaussian random variable which is F(B h )-measurable where F(B h ) denoes he σ-field generaed by B h. Case of he inegral in he sense of whie noise heory The reader who is no familiar wih he inegral wih respec o fbm in he sense of whie noise heory (also called fracional Wic-Iô inegral) may refer o Subsecion 4.1 or [7, 12]. We denoe his inegral Yi d B H, wih a similar noaion for he inegral w.r.. mbm. Denoe (W H ) [,T ] he fracional whie noise process (see Secion 4 and references herein for more deails). Se, for n N S n := Y d B hn. One compues: S n = q n 1 = x (n) 1 = Y ( q n 1 = Y d B h(x(n) ) = q n 1 = ( B h( ) 1 B h(x(n) where denoes he Wic produc. ) x (n) 1 Y )) = Y ( q n 1 = W h(x(n) ) d = q n 1 ( B h( ) 1 1 = Y ( W h(x(n) ) d ) B h(x(n) ) )), (3.7) 1

12 In he proof of Proposiion 3.2 we have shown ha he sequence ( q n 1 ( h(x = B (n) ) B h(x(n) 1 converges, in L 2 (Ω), o B1 h (, h()) d. By coninuiy of he Wic produc, we ge: lim S n = Y B1 h Y h () B h () B (, h()) d = Y B1 h h () Y ) )) n N B (, h()) d, (3.8) where he limi holds in L 2 (Ω). Now, for he Wic-Iô inegral w.r.. mbm defined in [18], one has: Y d B h = Y B h 1 = Y B h 1. Formula (3.8) hen reads Y B1 h 1 = lim Y d B hn h () Y B (, h()) d. We see ha, for his inegraion mehod, Formula (3.3) should be modified ino Y d B h := lim Y d B hn h () Y B (, h()) d. (3.9) The naural spaces of whie noise heory are he spaces (S p ), p being a posiive ineger. Equaliy (3.9) will be used in Secion 4 o define he inegral of an (S p )-valued process Y := (Y ) [,1] wih respec o mbm, where he limi and he las inegral (which is in he sense of Bochner) will hold in (S q ) for some ineger q, assuming hey boh exis. Noe ha he previous equaliy will also hold in (S q ). Case of he inegral in he sense of Sorohod We denoe his inegral wih he symbol δ. Thus, for insance, X δbh denoes he Sorohod inegral of he process X w.r.. B H. Coninuing wih our example Y = Y, where Y is a cenred Gaussian random variable, we use Theorem 7.4 in [16, secion 7], ha yields he general form of a Sorohod inegral wih respec o a Gaussian process. In our very simple case, his reads: Y δb h = Y 1 δb h = Y B h 1. Besides, [19, Proposiion 8] and [6, Theorem 6.2] yield ha Y δb H = Y d B H, as soon as Y δb H is defined. Thus, wriing T n := Y δb hn, we have T n = S n for all n (recall (3.7)). This promps us o defining he Sorohod inegral w.r.. o mbm again wih a Wic produc, i.e., using a formula analogous o (3.9): Y δb h := lim Y δb hn bu where he equaliy and limi would now hold in L 2 (Ω). h () Y B (, h()) d, (3.1) An advanage of hese definiions is ha hey will ensure, by consrucion, he equaliy Y d B h = Y δb h as soon as Y is inegrable w.r.. mbm in he sense of Sorohod. Case of pahwise inegrals In he case of he pahwise fracional inegral in he sense of [28], denoed X db h, he use of Formula (22) in [28] wih g an mbm and f = Y yields Y db h = Y B h 1. Thus he correc way o define our inegral w.r.. mbm in his case is o use a sandard produc, i.e. o se: Y db h := lim Y db hn h B () Y (, h()) d. (3.11) 11

13 3.2 Inegral wih respec o mbm hrough approximaing fbms We now define in a precise way our inegral wih respec o mbm. Le (E, E ) and (F, F ) be wo normed linear spaces, endowed wih heir Borel σ-field B(E) and B(F ). Le Y := (Y ) [,1] be an E-valued process (i.e Y belongs o E for every real in [, 1] and Y is measurable from (, 1) o (E, B(E)) ). Fix an inegraion mehod (M). As explained in he previous subsecion, we wish o define he inegral w.r.. an mbm B h in he sense of (M) by a formula of he ind: Y d (M) B h := lim Y d (M) B hn h () Y B (, h()) d, (3.12) where he meaning of he limi depends on (M) and where denoes he ordinary produc (in he case of pahwise inegrals) or Wic produc (in oher cases) depending on (M). For his formula o mae sense, i is cerainly necessary ha Y be (M) inegrable w.r.. fbm of all exponens α in h([, 1]). We hus define, for α (, 1), and H α E := { Y E [,1] : [,1] Y d (M) B α exiss and belongs o F }, H E = α h([,1]) We will always assume ha here exiss a subse Λ E of H E (maybe equal o H E ) which may be endowed wih a norm ΛE such ha (Λ E, ΛE ) is complee and which saisfies he following propery: here exiss M > and a real χ such ha for all pariions of [, 1] in inervals A 1,..., A n of equal size 1 n, Y.1 A1 ΛE Y.1 An ΛE M n χ Y ΛE. (3.13) When Y belongs o Λ E, Definiion (3.12) will be a valid one as soon as he limi and he las erm on he righ hand side exis. I urns ou ha a simple sufficien condiion guaranees he exisence of he limi of he inegral w.r.. lumped fbms. Define, for n N, he map L n : Λ E F Y [,1] H α E. Y d (M) B hn. (3.14) The following heorem provides a sufficien condiion under which (L n (Y )) n N converges in F. Theorem 3.3. Le (a n ) n N be an increasing sequence of posiive inegers such ha 2 n a n 1 2 2n for every n in N and such ha lim (n(a n 1)) ( a ) 1 =. Choose he sequence (q n ) n N n 1 used in (3.4) such ha q = 1 and q n1 = a n q n for all n in N. Assume ha he funcion I : Λ E (, 1) F defined by Y Λ E, α (, 1), I(Y, α) := Y d (M) B α, is θ-hölder coninuous wih respec o α uniformly in Y for a real number θ > χ, i.e. here exiss K > such ha Y Λ E, (α, α ) (, 1) 2, I(Y, α) I(Y, α ) F K α α θ Y ΛE. (3.15) Then he sequence of funcions (L n ) n N defined in (3.14) converges poinwise o a funcion L : Λ E F. Proof: For he sae of simpliciy, we will esablish he resul in he case where a n 2 which obviously fulfils he required condiions. The general case is similar. For n N and Y Λ E, L n (Y ) may be decomposed as L n (Y ) = 2 n 1 ( = 2 [ 2 n1, 21 Y d (M) B h( 2 n ) 2 n1 ) [ 21 2 n1, 22 Y d (M) B h( 2 n )) 2 n1 ). Now [,1] 12

14 L n1 (Y ) = 2 n 1 ( = 2 [ 2 n1, 21 Using assumpions (3.13) and (3.15), one obains 2 2 n1 ) Y d (M) B h( 2 n1 ) [ 21 2 n1, 22 Y d (M) B h( 2 n1 ) 21 2 n1 ) ). L n (Y ) L n1 (Y ) F = 2 n 1 ( ( = I Y.1 [ ) 21, h( 2 n1, 1 2 n 2 ) ) I ( Y.1 [ n n 1 = K 2 n 1 = ( I ( Y.1 [ ) 21 1, h( 2n1, 2 n Y.1 [(21).2 (n1),(1).2 n ) K 2 θ(n1) sup [,1] h () θ 2 n 1 = K M 2 θ(n1) n1, 1 ), h( 21 2 n 2 ) )) n1 F 2 ) ) I ( Y.1 [ ) n 21 1, h( 21 2n1, 2 n 2 ) )) n1 F h((2 1).2 (n1) ) h(.2 n ) θ ΛE Y.1 [(21).2 (n1),(1).2 n ) ΛE sup h () θ 2 nχ Y ΛE. (3.16) [,1] I follows ha he series n N (L n1(y ) L n (Y )) converges absoluely for any fixed Y Λ E, and consequenly (L n (Y )) n N converges o a limi L(Y ) as n goes o infiniy. For a process Y in Λ E, we will say ha h () Y B (, h()) is inegrable on [, 1] if h ()Y () B (, h()) is almos surely Riemann inegrable and h ()Y () B (, h()) d belongs o L 2 (Ω) in he case of he Sorohod inegral, h () Y B (, h()) d exiss in he sense of Bochner and belongs o F in he case of he Wic-Iô inegral (in his siuaion, L 2 (Ω) F ). The reader who is no familiar wih he Bochner inegral may refer o Secion 4 below and references herein, [,1] h B () Y (, h()) d exiss for almos every ω in he case of a pahwise inegral. We are finally able o define our inegral: Definiion 3.1 (Inegral wih respec o mbm in he sense of (M)). Assume ha Mehod (M) fulfils condiion (3.15) and le Y := (Y ) [,1] be an elemen of Λ E such ha he map h () Y B (, h() is inegrable. The inegral of Y wih respec o B h in he sense of (M) is defined as: Y d (M) B h := lim where he limi and equaliy boh hold in F. Y d (M) B hn h () Y B (, h()) d, (3.17) Remar 4. (i) Conrary o wha one migh expec in view of Proposiion 3.2, he second erm on he righ-hand side of (3.17) does no only depend on he choice of he fracional field B bu also on he mehod (M). Of course, he same is rue of he firs erm on he righ-hand side of (3.17). (ii) The main advanage of he above definiion is ha any nown sochasic inegral wih respec o fbm (e.g. pahwise inegrals, Sorohod or Wic-Iô inegral) gives rise o a corresponding sochasic inegral wih respec o mbm. (iii) Once again, noe ha E is no necessary a space of random variables (e.g E := (S p ) for some posiive ineger p; see Secion 4 below) and ha E may be differen from F (his will be he case in secion 4). Secions 4, 5 and 6 provide hree examples of applicaion of Theorem

15 4 Wic-Iô inegral wih respec o mbm hrough approximaing fbms Our aim in his secion is o consruc a Wic-Iô inegral w.r.. mbm using approximaing fbms. A direc approach o Wic-Iô inegraion w.r.. mbm is presened in [18], where Iô and Tanaa formulas are also obained. An applicaion of his inegral in mahemaical finance may be found in [1]. We shall compare he inegral obained hrough approximaing fbms wih he direc approach of [18] in Subsecion 4.4. For definieness, we will use he field B 1 (as in [18]), bu any oher field would lead o similar developmens. We firs briefly recall some basic facs abou whie noise heory and he Bochner inegral, as well as on he consrucion of he inegral w.r.. fbm in he spiri of [5, 6, 7, 12]. 4.1 Recalls on whie noise heory and he Bochner inegral Whie noise Theory Define he measurable space (Ω, F) by seing Ω := S (R) and F := B(S (R)), where B denoes he σ-algebra of Borel ses. There exiss a unique probabiliy measure µ on (Ω, F) such ha, for every f in L 2 (R), he map <., f >: Ω R defined by <., f > (ω) =< ω, f > (where < ω, f > is by definiion ω(f), i.e. he acion of he disribuion ω on he funcion f) is a cenred Gaussian random variable wih variance equal o f 2 L 2 (R) under µ. For every n in N, define e n(x) := ( 1) n π 1/4 (2 n n!) 1/2 e x2 /2 d n dx (e x2 ) he n h Hermie funcion. Le ( n p ) p Z be he family norms defined by f 2 p := = (2 2)2p < f, e > 2 L (R), for all (p, f) in Z L 2 (R). The operaor A defined 2 on S (R) by A := d2 dx x 2 1 admis he sequence (e 2 n ) n N as eigenfuncions and he sequence (2n 2) n N as eigenvalues. We denoe (L 2 ) he space L 2 (Ω, G, µ) where G is he σ-field generaed by (<., f >) f L 2 (R). For every random variable Φ of (L 2 ) here exiss, according o he Wiener-Iô heorem, a unique sequence (f n ) n N of funcions f n in L 2 (R n ) such ha Φ can be decomposed as Φ = n= I n (f n ), where L 2 (R n ) denoes he se of all symmeric funcions f in L 2 (R n ) and I n (f) denoes he n h muliple Wiener-Iô inegral of f wih he convenion ha I (f ) = f for consans f. Moreover he equaliy E[Φ 2 ] = n= n! f n 2 L 2 (R n ) holds, where E denoes he expecaion wih respec o µ. For any Φ := n= I n (f n ) saisfying he condiion n= n! A n f n 2 <, define he elemen Γ(A)(Φ) of (L2 ) by Γ(A)(Φ) := n= I n(a n f n ), where A n denoes he n h ensor power of he operaor A (see [16, Appendix E] for more deails abou ensor producs of operaors). The operaor Γ(A) is densely defined on (L 2 ). I is inverible and is inverse Γ(A) 1 is bounded. Le, for ϕ in (L 2 ), ϕ 2 := ϕ 2 (L 2 ) and, for n in N, le Dom(Γ(A) n ) be he domain of he n h ieraion of Γ(A). Define he family of norms ( p ) p Z by: Φ p := Γ(A) p Φ = Γ(A) p Φ (L 2 ), p Z, Φ (L2 ) Dom(Γ(A) p ). For p in N, define (S p ) := {Φ (L 2 ) : Γ(A) p Φ exiss and belongs o (L 2 )} and define (S p ) as he compleion of he space (L 2 ) wih respec o he norm p. As in [17], we le (S) denoe he projecive limi of he sequence ((S p )) p N and (S) he inducive limi of he sequence ((S p )) p N. This means ha we have he equaliies (S) = (S p ) (resp. (S) = (S p )) and ha convergence in (S) (resp. in p N p N (S) ) means convergence in (S p ) for every p in N (resp. convergence in (S p ) for some p in N ). The space (S) is called he space of sochasic es funcions and (S) he space of Hida disribuions. One can show ha, for any p in N, he dual space (S p ) of S p is (S p ). Thus we will wrie (S p ), in he sequel, o denoe he space (S p ). Noe also ha (S) is he dual space of (S). We will noe <, > he dualiy brace beween (S) and (S). If φ, Φ belong o (L 2 ) hen we have he equaliy <Φ, ϕ > = < Φ, ϕ > (L2 ) = E[Φ ϕ]. A funcion Φ : R (S) is called a sochasic disribuion process, or an (S) process, or a Hida process. A Hida process Φ is said o be differeniable a R if lim r r 1 (Φ( r) Φ( )) exiss in (S). 14

16 For f in L 2 (R), we define he Wic exponenial of <., f >, noed : e <.,f> :, as he (L 2 ) random variable equal o e <.,f> 1 2 f 2. The S-ransform of an elemen Φ of (S ), noed S(Φ), is defined as he funcion from S (R) o R given by S(Φ)(η) := <Φ, : e <.,η> : > for every η in S (R). Finally for every (Φ, Ψ) (S) (S), here exiss a unique elemen of (S), called he Wic produc of Φ and Ψ and noed Φ Ψ, such ha S(Φ Ψ)(η) = S(Φ)(η) S(Ψ)(η) for every η in S (R) Fracional and mulifracional Whie noise We inroduce wo operaors, denoed M H and M H, ha will prove useful for he definiion of he inegral wih respec o fbm and mbm. Operaors M H and M H Le H be a fixed real in (, 1). Following [12] and references herein, define he operaor M H, specified in he Fourier domain, by M H (u)(y) := 2π c H y 1/2 H û(y) for every y in R. This operaor is well defined on he homogeneous Sobolev space of order 1/2 H, denoed L 2 H (R) and defined by L2 H (R) := {u S (R) : û = T f ; f L 1 loc (R) and u H < }, where he norm H derives from he inner produc <, > H defined on L 2 H (R) by < u, v > H := 1 c 2 R ξ 1 2H u (ξ) v (ξ)dξ and where c H was given in he H definiion of he fracional field B 1 in Secion 1. The definiion of he operaor M H is quie similar. More precisely, define for every H in (, 1), he space Γ H (R) := {u S (R) : û = T f ; f L 1 loc (R) and u δ H (R) < }, where he norm δ H (R) derives from he inner produc on Γ H (R) defined by < u, v > δh := 1 c 2 R (β H ln ξ ) 2 ξ 1 2H u (ξ) v (ξ) dξ. H Following [18], define he operaor M H from (Γ H(R), <, > δh (R)) o (L 2 (R), <, > L (R)), in he Fourier 2 domain, by: ÿ M H (u)(y) := (β H ln y ) 2π c H y 1/2 H û(y), for every y in R. The reader ineresed in he properies of M H and M H may refer o [18, Secions 2.2 and 4.2]. Fracional and mulifracional Whie noise Recall he following resul ([18, (5.1)]): Almos surely, for every, B h = B 1 (, h()) = <., M h() (1 [,] ) > = = ( M h()(e )(s) ds ) <., e >. (4.1) We now define he derivaive in he sense of (S) of mbm. Define he (S )-valued process W h := (W h ) [,1] by W h := = [ d ( d M h()(e )(s) ds )] <., e >. (4.2) Theorem-Definiion 4.1. [18, Theorem-definiion 5.1] The process W h defined by (4.2) is an (S) - process which verifies, in (S), he following equaliy: W h = = M h() (e )() <., e > h () ( = M H (e )(s) H=h() ds ) <., e >. (4.3) Moreover he process B h is (S) -differeniable on [, 1] and verifies dbh d () = W h in (S). When he funcion h is consan and idenically equal o H, we will wrie W H := (W H ) [,1] and call he (S) -process W H a fracional whie noise. Noe ha (4.3) may be wrien as W h = W h() h () B1 (, h()), (4.4) where W h() is nohing bu W H H=h() and where he equaliy holds in (S). 15

17 4.1.3 Bochner inegral Since he objecs we are dealing wih are no longer random variables in general, he Riemann or Lebesgue inegrals are no relevan here. However, aing advanage of he fac ha we are woring wih vecor linear spaces, we may use Peis or Bochner inegrals. In he frame of he Wic-Iô inegral, he space E, defined a he beginning of Secion 3.2 will be a space (S p ) for some ineger p. The fac ha we need a norm on H E suggess he use of Bochner inegral. A nice survey of his opic may be found in [17, p.247]. We only recall here he definiion and wo basic resuls. Definiion 4.1 (Bochner inegral [17], p.247). Le I be a subse of [, 1] endowed wih he Lebesgue measure. One says ha Φ : I (S) is Bochner inegrable of index p on I if i saisfies he wo following condiions: 1. Φ is wealy measurable on I, i.e. < Φ, ϕ > is measurable on I for every ϕ in (S). 2. There exiss p in N such ha Φ (S p ) for almos every in I and such ha Φ p belongs o L 1 (I, d). The Bochner-inegral of Φ on I is denoed I Φ d. Proposiion 4.1. If Φ: I (S) is Bochner-inegrable on I wih index p hen I Φ d p I Theorem 4.2 ([17], Theorem 13.5). Le Φ := (Φ ) [,1] be an (S) -valued process such ha: (i) S(Φ )(η) is measurable for every η in S (R). Φ p d. (ii) There exis p in N, b in R and a funcion L in L 1 ([, 1], d) such ha, for a.e. in [, 1], S(Φ )(η) L() e b η 2 p, for every η in S (R). Then Φ is Bochner inegrable on [, 1] and Φ(s) ds (S q) for every q > p such ha 2be 2 D(q p) < 1 where e denoes he base of he naural logarihm and where D(r) := 1 2 2r n=1 1 n for r in (1/2, ). 2r 4.2 Wic-Iô inegral wih respec o fbm The fracional Wic-Iô inegral wih respec o fbm (or inegral w.r.. fbm in he whie noise) was inroduced in [12] and exended in [5] using he Peis inegral. However, in order o use Theorem 3.3, we need ha Y s belongs o (S p ) for almos every real s. I hen seems reasonable o assume ha (Y s ) s [,1] is Bochner inegrable on [, 1]. For his reason, we now paricularize he fracional Wic-Iô inegral wih respec o fbm of [12] and [5] in he framewor of he Bochner inegral. Definiion 4.2 (Wic-Iô inegral w.r. fbm in he Bochner sense). Le H (, 1), I be a Borel subse of [, 1], B H := (B H ) I be a fracional Brownian moion of Hurs index H, and Y := (Y ) I be an (S) -valued process verifying: (i) here exiss p N such ha Y (S p ) for almos every I, (ii) he process Y W H is Bochner inegrable on I. Then, Y is said o be Bochner-inegrable wih respec o fbm on I and is inegral is defined by: Y s d Bs H := Y s Ws H ds. (4.5) I Lemma 4.3. Le Y := (Y ) [,1] be an (S) -valued process, Bochner inegrable of index p N. Then Y is inegrable on [, 1], wih respec o fbm of any Hurs index H, in he Bochner sense. Moreover, for any H in (, 1), [,1] Y s d B H s belongs o (S r ) for every r p 1 if p 2 and for every r p 2 if p {, 1}. I 16

18 Proof: Fix H (, 1), p 2 and r p 1. The map Y W H is wealy measurable since S(Y W H )(η) is measurable for all η S (R). Using [17, Remar 2 p.92], one obains ha, for almos all in [, 1], Y W H r Y p W H p <. Hence Y W H belongs o (S r ). Since he map W H r is coninuous for all ineger r 2 (see [18, Proposiion 5.9]), one also ges: Y W H r d ( sup W H ) p Y p d <. [,1] This shows ha Y W H is Bochner-inegrable of index r. Le us now assume ha p {, 1}. I is sufficien o chec ha Theorem 4.2 applies. Condiion (i) is obviously fulfilled. Moreover, using [17, p.79], we obain ha, for every (, η) in [, 1] S (R), S(Y W H )(η) Y p e 1 2 η 2 2 sup W H 2 =: L() e 1 2 η 2 2. [,1] Since Y is Bochner inegrable of index p, i is clear ha L belongs o L 1 ([, 1], d). Moreover, e 2 D(r p ) < 1, for every r p 2. Theorem 4.2 hen allows o conclude ha Y W H is Bochner inegrable of index r. The following lemma, he proof of which is obvious in view of hypohesis (H 2 ), will be useful in he proof of Proposiion 4.7 below. Lemma 4.4. For every p in N, he map (, H) B1 (, H) is coninuous from [, 1] ino ((S p), p ). In paricular, for every subse [a, b] of (, 1), here exiss a posiive real κ such ha: p N, sup (s,h) [,] [a,b] B 1 (s, H) p κ. (4.6) 4.3 Sochasic inegral wih respec o mbm wih approximaing fbms We consruc in his secion he Wic-Iô inegral w.r.. mbm using approximaing fbms. In ha view, we shall apply Theorem 3.3. Fix (p, s ) in N 2 such ha s max{p 1, 3}. Se E := (S p ) and F := (S s ). By definiion we have H E = { (Y ) [,1] (S p ) R : [,1] Y d B α (S s ), α h([, 1]) }. Define also he se Λ E := (Y ) [,1] (S p ) R : Y is Bochner inegrable of index p on [, 1] equipped wih he norm Φ ΛE := Φ p d. The inclusion Λ E H E resuls from Lemma 4.3 whereas he fac ha (Λ E, ΛE ) is complee is a sraighforward consequence of [14, Theorem p.82]. Moreover, ΛE fulfils condiion (3.13) wih χ =. Lemma 4.5. The Wic-Iô inegral w.r.. fbm verifies condiion (3.15) wih θ = 1. Proof: Since (S p ) (S 2 ) if p belongs o {; 1}, we assume from now ha p 2 and ha s p 1. Le Y Λ E. Lemma 4.3 enails ha Y is inegrable wih respec o fbm in he Bochner sense, for all α in (, 1) and ha I(Y, α) = [,1] Y d B α belongs o (S s ). Now for all (α, α ) in (, 1) 2, using he same argumens we used in he proof of Lemma 4.3, and hus I(Y, α) I(Y, α ) s = [,1] Y (W α W α ) d s Y p W α W α p d ( sup W α ) W α p Y ΛE, [,1] sup I(Y, α) I(Y, α ) s sup W α W α p. (4.7) Y ΛE 1 [,1] By definiion, W α W α 2 p = (M α(e )() M α (e )()) 2 =. (22) 2p For all (, ) in [, 1] N, he funcion α M α (e )() is differeniable on (, 1) (his is Lemma 5.5 in [18]). Using poin 1 of [18, lemma 5.6] and he mean value heorem, one obains he following fac: for all [a, b] (, 1), here exiss a posiive real ρ such ha for all (, α, α, ) [, 1] [a, b] 2 N: M α (e )() M α (e )() ρ ( 1) 2/3 ln( 1) α α. 17

19 As a consequence, we ge W α W α 2 p ρ 2 α α 2 (1) 4/3 ln 2 (1) = 2 2p (1) 2p. Wih (4.7), one obains sup I(Y, α) I(Y, α ) s α α γ p, (4.8) Y ΛE 1 where γ p := ρ Ä ä ln 2 1/2 =1 2(p is finie since p 2. 2/3) A consequence of he previous lemma is ha Theorem 3.3 applies, ha is, lim [,1] Y d B hn as an elemen of (S s ). exiss Lemma 4.6. For every process Y := (Y ) [,1] Bochner-inegrable on [, 1], he map h () Y B 1 (, h()) is inegrable. Proof: Fix p 2 and s p 1. Using he same argumens as in he proof of Lemma 4.3 one easily prove ha h () Y B1 (, h()) on [, 1] is wealy measurable. Lemma 4.4 enails ha, for every p, sup B1 (s, H) p κ, (s,h) [,1] h([,1]) B 1 We hence ge h () Y B1 (, h()) ( s Y s p sup h (s) ) sup (s, h(s)) p <. s [,1] s [,1] Thus here exiss δ R, such ha h (s) Y s B1 (s, h(s)) s ds δ Y s p ds <. As a consequence, (, h())d is well defined in he sense of Bochner. h () Y B1 As a consequence of Lemmas 4.5 and 4.6, he inegral w.r.. mbm exiss as a limi of inegrals w.r.. fbms: Corollary 4.7. Le Y Λ E. Then Y d B h := lim Y d B hn h () Y B1 (, h()) d, (4.9) where he limi and he equaliy hold in (S s ), is well-defined and belongs o (S s ). 4.4 A comparison beween mulifracional Wic-Iô inegral and limiing fracional Wic-Iô inegral A mulifracional Wic-Iô inegral wih respec o mbm was defined in [18]. I is ineresing o chec wheher i coincides wih he one defined in Corollary 4.7. In ha view, we need o adap he definiion of [18], which used Peis inegrals, o deal wih Bochner inegrals. Definiion 4.3 (Mulifracional Wic-Iô inegral in Bochner sense). Le I be a Borelian conneced subse of [, 1], B h := (B h ) I be a mulifracional Brownian moion and Y := (Y ) I be a (S) -valued process such ha: (i) There exiss p N such ha Y (S p ) for almos every I, (ii) he process Y W h is Bochner inegrable on I. Y is hen said o be inegrable on I wih respec o mbm in he Bochner sense or o admi a mulifracional Wic-Iô inegral. This inegral is defined o be I Y s Ws h ds. Remar 5. From he definiion of ( ) W h [18, proposiion 5.9], and he proof of Lemma 4.3, i [,1] is clear ha every (S) -valued process Y := (Y ) I which is Bochner inegrable on I of index p, is inegrable on I wih respec o mbm, in he Bochner sense. Moreover [,1] Y db h belongs o (S r ), where r was defined in Lemma 4.3. In order o compare our wo inegrals wih respec o mbm when hey boh exis, i seems naural o assume ha Y = (Y ) [,1] is a Bochner inegrable process of index p N. The space E and he norm ΛE are defined as in he previous subsecion. 18

20 Theorem 4.8. Le Y = (Y ) [,1] be a Bochner inegrable process of index p N. Then Y is inegrable wih respec o mbm in boh senses of Corollary 4.7 and Definiion 4.3. Moreover [,1] Y d B h and I Y s Ws h ds are equal in (S ). Proof: Since Y is a Bochner inegrable process of index p N, Proposiion 4.7 and Remar 5 enail ha [,1] Y s d B h s exiss in (S s ) and ha I Y s W h s ds exis in (S r ), where s has been defined jus above Proposiion 4.7 and r has been defined in Lemma 4.3. Moreover, hans o (4.9) and using (3.4) and (4.5), we may wrie, in (S s ), [,1] Y s d B h s = lim = lim = lim [,1] Y d B hn qn 1 = qn 1 = [,1] h () Y B1 (, h()) d 1 (n) [x,x(n) 1 (n) [x,x(n) )() Y d B h( ) 1 )() Y W h( ) 1 Besides, Definiion 4.3 and (4.4) enail ha, in (S r ), Y W h d = Y W h() d h () Y B1 (, h()) d d h () Y B1 (, h()) d. (4.1) Since s r we have (S r ) (S s ). Thus i remains o show ha, in (S s ), h () Y B 1 (, h()) d. (4.11) L(Y ) := lim qn 1 = 1 (n) [x,x(n) )() Y W h( ) 1 d is equal o M(Y ) := Y W h() d. Since L(Y ) and M(Y ) boh belong o (S s ), i is sufficien o show ha hey have he same S-ransform. Using [17, Theorem 8.6], one has, for η in S (R), S(L(Y ))(η) = lim S( q n 1 = Using now (ii) of [18, Theorem 5.12], one ges S(L(Y ))(η) = lim = lim qn 1 = x (n) 1 [,1] S(Y )(η) 1 (n) [x,x(n) )() Y W h(x 1 S(Y )(η) S(W h(x(n) )(η) d Å qn 1 ) = 1 [ )() M,x(n) 1 h (n) ) ( d ) (η). ) (η)() ã d. (4.12) The map (, H) M H (η)() is coninuous (see [18, Lemma 5.5]). Define K η := For all n in N and in [, 1], one has S(Y )(η) q n 1 = 1 [,x(n) sup M H (η)(). (,H) [,1] h([,1]) )() M h( )(η)() Kη e 1 2 η 2 p Y p. (4.13) 1 The map K η e η 2 p Y p belongs o L 1 (R, d). In addiion, for almos every in [, 1], lim qn 1 = 1 [,x(n) )() M 1 h ( Thus, by dominaed convergence and using (4.13), one ges, ) (η)() = Mh() (η)(), S(L(Y ))(η) = [,1] S(Y )(η) M h() (η)() d = [,1] S(Y )(η) S(W h() )(η) d = S ( [,1] Y W h() d ) (η), Since he map S : Φ S(Φ) from (S) ino iself is injecive, one deduces ha L(Y ) = [,1] Y W h() and he proof is complee. d, 19