From Stochastic Integration wrt Fractional Brownian Motion to Stochastic Integration wrt Multifractional Brownian Motion
aa r X i v : . [ m a t h . P R ] M a y From Stochastic Integral w.r.t.
FractionalBrownian Motion to Stochastic Integral w.r.t.
Multifractional Brownian Motion
To appear in Annals of the University of Bucarest
Joachim Lebovits
Abstract -
Because of numerous applications e.g. in finance and in In-ternet traffic modelling, stochastic integration w.r.t. fractional Brownianmotion (fBm) became a very popular topic in recent years. However, sincefBm is not a semi-martingale the Itô integration can not be used for integra-tion w.r.t. fBm and one then needs specific developments. MultifractionalBrownian motion (mBm) is a Gaussian process that generalizes fBm byletting the local Hölder exponent vary in time. In addition to the fieldsmentioned above, it is useful in many and various areas such as geology andbiomedicine. In this work we start from the fact, established in [9, Thm2.1.(i)], that an mBm may be approximated, in law, by a sequence of “tan-gent" fBms. We used this result to show how one can define a stochasticintegral w.r.t. mBm from the stochastic integral w.r.t. fBm, defined in [4],in the white noise theory sense.
Key words and phrases :
Fractional and multifractional Brownian mo-tions, Gaussian processes, convergence in law, white noise theory, Wick-Itôintegral.
Mathematics Subject Classification (2010) : Fractional Brownian motion (fBm) is a centred Gaussian process thathas been used a lot, and still is, these recent years to model many, naturalor artificial, phenomena such as physics, geophysics, financial and Internettraffic modeling, image analysis and synthesis and more. One of the advan-tages, that makes the use of fBm so popular to model phenomena, is the factthat fBm is a long range dependence process, that has the ability to matchany prescribed constant local regularity. For any H in (0 , , its covariancefunction R H reads: R H ( t, s ) := γ H | t | H + | s | H − | t − s | H ) , where γ H is a positive constant. The parameter H is called the Hurst index.Besides, when H = , fBm reduces to standard Brownian motion. Various1integral representations of fBm are known, including the harmonizable andmoving average ones [13], as well as representations by integrals over a finitedomain [1, 7].The fact the Hölder exponent of a fBm remains constant and the same allalong its trajectory restricts its application in some situations and moreimportantly does not seem to be adapted to describe or to model phenomenathat present in the same time a long range dependance (which requires H > / ) and irregular trajectories (which requires H < / ). MultifractionalBrownian motion was introduced to overcome these limitations. The basicidea is to replace the real H by a function t h ( t ) , ranging in (0 , .Several definitions of multifractional Brownian motion exist. The firstones were proposed in [12] and in [3]. They then have been extended in[14]. Finally, a more general definition of mBm, that includes all the defi-nitions given in [12, 3, 14], has been introduced in [9]. In the present work,and for sake of simplicity, we will only deal with the harmonizable versionof mBm, which we will call “the” mBm.More precisely, a deterministic function h : R → (0 , being fixed, the mBmof functional parameter h , noted B h := ( B ht ) t ∈ R , is defined by: B ht := 1 c h ( t ) Z R e itu − | u | h ( t )+1 / f W ( du ) , (1.1)where c x := Ä πx )Γ(2 − x ) x (1 − x ) ä , for every x in (0 , , and where f W denotesa complex-valued Gaussian measure. Denote R h the covariance function of B h . Thanks to [2], we know that: R h ( t, s ) = c ht,s c h ( t ) c h ( s ) î Ä | t | h t,s + | s | h t,s − | t − s | h t,s äó , where h t,s := h ( t )+ h ( s )2 .A word on notation: B H. or B h ( t ) . will always denote an fBm with Hurstindex H or h ( t ) , while B h. will stand for an mBm. Note that B ht := B h ( t ) t ,for every real t . Besides, and in order to simplify some notations in thesequel, denote B ( t, H ) := c H R R e itu − | u | H +1 / f W ( du ) , for every H in (0 , . Outline of the paper
The remaining of this paper is organized as follows. We explain in the firstpart of Section how an mBm can be approximated in law by a sequence of“tangent” fBms. The second part of Section is devoted to some heuristicsabout the way to define a stochastic integral w.r.t. mBm as a limit of integralsw.r.t. approximating fBms. Section 3 provides the background on whitenoise theory and on fractional white noise that will allow us to define, in arigorous manner, the limiting fractional Wick-Itô integral in Section 4. Themain result of Section 4 being Theorem 4.1. Finally, we compare in Section5 the limiting fractional Wick-Itô integral to the stochastic integral withrespect to mBm that has been defined in [11]. We start this section with the following result, that will be useful in allthis paper. Recalling the definition of B given at the end of the previoussection. Proposition 2.1
For every [ a, b ] × [ c, d ] in R × (0 , , there exists Λ in R ∗ + such that for every ( t, s, H, H ′ ) in [ a, b ] × [ c, d ] , E [( B ( t, H ) − B ( s, H ′ )) ] ≤ Λ Ä | t − s | c + | H − H ′ | ä , where E [ Y ] denotes the expectation of a real random variable Y , which be-longs to L (Ω , F , P ) . Proof.
Let ( s, t, H, H ′ ) be fixed in [ a, b ] × [ c, d ] . Since, for every H in [ c, d ] ,the process { B ( t, H ) , t ∈ R } is a fractional Brownian motion of Hurst index H , we know, using the triangular inequality, that it is sufficient to show that I H,H ′ t := E [( B ( t, H ) − B ( t, H ′ )) ] ≤ Λ | H − H ′ | . One has: I H,H ′ t = Z R | e itξ − ξ | (cid:12)(cid:12)(cid:12) c H | ξ | / − H − c H ′ | ξ | / − H ′ (cid:12)(cid:12)(cid:12) dξ. For ξ in R ∗ , the map f ξ : [ c, d ] → R + , defined by f ξ ( H ) := c H | ξ | / − H is C , since H c H is C ∞ on (0 , . Thus there exists a positive real D suchthat, for all ( ξ, H ) in R ∗ × [ c, d ] , | f ′ ξ ( H ) | ≤ D | ξ | / − H (1 + | ln( | ξ | ) | ) ≤ D ( | ξ | / − c + | ξ | / − d ) (1 + | ln( | ξ | ) | ) . Thanks to the mean-value theorem, one can write I H,H ′ t ≤ C | H − H ′ | Z R | e itξ − | | ξ | ( | ξ | / − c + | ξ | / − d ) (1 + | ln( | ξ | ) | ) dξ ≤ C | H − H ′ | Ä Z | ξ | > | ξ | ) | ξ | c dξ + Z | ξ |≤ | ξ | − d (1 + | ln( | ξ | ) | ) dξ ä ≤ C | H − H ′ | , where C stands for a constant, independent of t and H , whose precise valueis unimportant and which may change from line to line. (cid:3) Remark 2.1
Since B := ( B ( t, H )) ( t,H ) ∈ R × [0 , is a Gaussian field, the pre-vious result, as well as Kolmogorov’s criterion, entail that the field B hasa d -Hölder continuous version for any d in (0 , ∧ c ) . In the sequel we willalways work with such a version. Since a mBm is generalization of fBm, it is a natural question to wonderif an mBm may be approximated by patching adequately chosen fBms; thesense of this approximation remaining to define. For notational simplicitywe take [ a, b ] := [0 , in the sequel. Heuristically, we divide [0 , into“small” intervals [ t i , t i +1 ) , and replace on each of these B h by the fBm B H i where H i = h ( t i ) . It seems reasonable to expect that the resulting process P i B H i t [ t i ,t i +1 ) ( t ) will converge, in law, to B h when the sizes of the intervals [ t i , t i +1 ) go to 0. The notations remains the same as previously. Let h :[0 , → (0 , be a conitnuous deterministic function and denote B h thefixed mBm of functional parameter h . Let us explain how this mBm canbe approximated on [0 , by patching together fractional Brownian motionsdefined on a sequence of partitions of [0 , .In that view, we choose an increasing sequence ( q n ) n ∈ N of integers suchthat q := 1 and n ≤ q n ≤ n for all n in N ∗ . For any n in N , define x ( n ) := { x ( n ) k ; k ∈ [[0 , q n ]] } where x ( n ) k := kq n for k in [[0 , q n ]] (for integers p and q with p < q , [[ p, q ]] denotes the set { p ; p + 1; · · · ; q } ). Define, for n in N ,the partition A n := { [ x ( n ) k , x ( n ) k +1 ); k ∈ [[0 , q n − } ∪ { x ( n ) q n } . It is clear that A := ( A n ) n ∈ N is a decreasing nested sequence of subdivisions of [ a, b ] ( i.e. A n +1 ⊂ A n , for every n in N ).For t in [0 , and n in N there exists a unique integer p in [[0 , q n − suchthat x ( n ) p ≤ t < x ( n ) p +1 . We will note x ( n ) t the real x ( n ) p in the sequel. It is clearthat the sequence ( x ( n ) t ) n ∈ N is increasing and converges to t as n tends to + ∞ . Besides, define for n in N , the function h n : [0 , → (0 , by setting h n (1) = h (1) and, for any t in [0 , , h n ( t ) := h ( x ( n ) t ) . The sequence of stepfunctions ( h n ) n ∈ N converges pointwise to h on [0 , . Define, for t in [0 , and n in N , the process B h n t := B ( t, h n ( t )) = q n − X k =0 [ x ( n ) k ,x ( n ) k +1 ) ( t ) B ( t, h ( x ( n ) k )) + { } ( t ) B (1 , h (1)) . (2.1)Note that, despite the notation, the process B h n is not an mBm, as h n isnot continuous.The following theorem, the proof of which can be found in [9, Thm 2.1.(i)],shows that mBm appears naturally as a limit object of sums of fBms. Theorem 2.1 (Approximation theorem)
Let A be a sequence of parti-tions of [0 , as defined above, and consider the sequence of processes definedin (2.1) .If h is β -Hölder continuous for some positive real β , then the sequence ofprocesses ( B h n ) n ∈ N converges, in law, to the process B h . In oher words, { B h n t ; t ∈ [ a, b ] } law −−−−−→ n → + ∞ { B ht ; t ∈ [ a, b ] } . In the remaining of this paper, we consider a C deterministic function h : R → (0 , . Moreover B still denotes the Gaussian field defined at theend of Section 1 and B h the mBm defined by (1.1). The mBm B h thenverifies B ht a.s. := B ( t, h ( t )) , for every t .The result of Theorem 2.1 suggests that one may define stochastic integralswith respect to mBm as limits of integrals with respect to approximatingfBms. Let first notice the following fact, the proof of which can be foundin [9, Appendix B], that will be useful to define the integral with respect tomBm. Proposition 2.2
For every real t , the map H B ( t, H ) , from (0 , to L (Ω) , is C . Moreover, it fulfills the following Hölder condition. For all [ a, b ] × [ c, d ] ⊂ R × (0 , there exists ∆ in R ∗ + such that: E î ( ∂ B ∂H ( t, H ) − ∂ B ∂H ( s, H ′ )) ó ≤ ∆ ( | t − s | + | H − H ′ | ) , where ∂ B ∂H ( t, H ) denotes, for every real t , the derivative with respect to H ofthe map H B ( t, H ) . Let us explain now, in a heuristic way, how to define an integral with respectto mBm using approximating fBms. Write the “differential” of B ( t, H ) : d B ( t, H ) = ∂ B ∂t ( t, H ) dt + ∂ B ∂H ( t, H ) dH. Of course, this is only formal as t B ( t, H ) is not differentiable in the L -sense nor almost surely with respect to t . It is, however, in the sense ofHida distributions (this will be made precise in Section 3, see in particular(3.2)). With a differentiable function h in place of H , this (again formally)yields d B ( t, h ( t )) = ∂ B ∂t ( t, h ( t )) dt + h ′ ( t ) ∂ B ∂H ( t, h ( t )) dt. (2.2)Of course the first term of the right hand side of (2.2) has no meaning a priori since mBm is not differentiable with respect to t . However, con-tinuing with our heuristic reasoning, we then approximate ∂ B ∂t ( t, h ( t )) by lim n → + ∞ P q n − k =0 [ x ( n ) k ,x ( n ) k +1 ) ( t ) ∂ B ∂t ( t, h n ( t )) . This formally yields: d B ( t, h ( t )) ≈ lim n → + ∞ P q n − k =0 [ x ( n ) k ,x ( n ) k +1 ) ( t ) ∂ B ∂t ( t, h n ( t )) dt + h ′ ( t ) ∂ B ∂H ( t, h ( t )) dt. (2.3)For the sake of notational simplicity, we will consider integrals over the inter-val [0 , . Let us note R [0 , Y t d ⋄ B Ht the integral of a process Y with respectto a fBm of Hurst index H , in the white noise theory sense (that will be fullydetailed in the next section), assuming it exists.From (2.3), and assuming we may exchange integrals and limits, we wouldthus like to define, for suitable processes Y , the integral with respect to themBm B h , noted R Y t d ⋄ B ht , by setting, R Y t d ⋄ B ht = lim n → + ∞ P q n − k =0 R x ( n ) k +1 x ( n ) k Y t d ⋄ B h ( x ( n ) k ) t + R Y t h ′ ( t ) ∂ B ∂H ( t, h ( t )) dt, (2.4)where the first term of the right-hand side of (2.4) is a limit, in a sense to bemade precise of a sum of integrals with respect to fBms and the second termis a weak integral (see Section 3). The following notation will be useful: Notation (integral with respect to lumped fBms)
Let Y := ( Y t ) t ∈ [0 , be a real-valued process on [0 , which is integrable with respect to all fBms ofindex H in h ([0 , in the white noise theory sense. We denote the integralwith respect to lumped fBms by: Z Y t d ⋄ B h n t := q n − X k =0 Z [ x ( n ) k ,x ( n ) k +1 ) ( t ) Y t d ⋄ B h ( x ( n ) k ) t , n ∈ N , (2.5)where ( q n ) n ∈ N and x ( n ) := { x ( n ) k ; k ∈ [[0 , q n ]] } have been defined in Section2.1. With this notation, our tentative definition of an integral w.r.t. to mBm(2.4) reads: Z Y t d ⋄ B ht = lim n → + ∞ Z Y t d ⋄ B h n t + Z Y t h ′ ( t ) ∂ B ∂H ( t, h ( t )) dt, The drawback with the previous definition is that, when R Y t h ′ ( t ) ∂ B ∂H ( t, h ( t )) dt will belong to L (Ω) , it will not be centred, a priori . For this reason and be-cause one can see the Wick product as a centered product, we would ratherchoose, as a definition of R Y t d ⋄ B ht , the following one: Z Y t d ⋄ B ht := lim n → + ∞ Z Y t d ⋄ B h n t + Z h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt, (2.6)where ⋄ denotes the Wick product (that will be rigorously defined in thenext section).In order to make the previous statement rigorous, we need to give a precisemeaning to the right hand side of (2.6). In particular, giving a precisemeaning to R Y t d ⋄ B Ht and thus to R Y t d ⋄ B h n t is crucial. This is the aimof the next section. This section is divided into two parts. In the first one we briefly recall somebasic facts about white noise theory and the Bochner integral. In the sec-ond part we particularize the definition of the Fractional Wick-Itô Integral,defined in [4, 5, 6, 8], into the framewok of Bochner integral.
The following subsection being on purpose extremely short. The reader whois no familiar with white noise theory should refer to [10] and referencestherein.Define the measurable space (Ω , F ) by setting Ω := S ′ ( R ) and F := B ( S ′ ( R )) , where B denotes the σ -algebra of Borel sets. Denotes µ theunique probability measure on (Ω , F ) such that, for every f in L ( R ) , themap < ., f > : Ω → R defined by < ., f > ( ω ) = < ω, f > (where <, > contin-uously in L ( R ) extends the action of tempered distributions on Schwartzfunctions) is a centred Gaussian random variable with variance equal to k f k L ( R ) under µ .We also denote ( L ) the space L (Ω , G , µ ) where G is the σ -field generated by ( < ., f > ) f ∈ L ( R ) , and for every n in N , define the n − th Hermite functionby e n ( x ) := ( − n π − / (2 n n !) − / e x / d n dx n ( e − x ) . Denote A the operatordefined on S ( R ) by A := − d dx + x + 1 and Γ( A ) the second quantizationoperator of A (see [10, Section 4.2]).Denote, for ϕ in ( L ) , k ϕ k := k ϕ k L ) and, for n in N , let D om (Γ( A ) n ) be the domain of the n − th iteration of Γ( A ) . Define the family of norms ( k k p ) p ∈ Z by: k Φ k p := k Γ( A ) p Φ k = k Γ( A ) p Φ k ( L ) , ∀ p ∈ Z , ∀ Φ ∈ ( L ) ∩ D om (Γ( A ) p ) . For p in N , define ( S p ) := { Φ ∈ ( L ) : Γ( A ) p Φ exists and belongs to ( L ) } and define ( S − p ) as the completion of the space ( L ) with respect to the norm k k − p . As in [10], we let ( S ) denote the projective limit of the sequence (( S p )) p ∈ N and ( S ) ∗ the inductive limit of the sequence (( S − p )) p ∈ N . Thismeans that we have the equalities ( S ) = ∩ p ∈ N ( S p ) (resp. ( S ) ∗ = ∪ p ∈ N ( S − p ) )and that convergence in ( S ) (resp. in ( S ) ∗ ) means convergence in ( S p ) forevery p in N (resp. convergence in ( S − p ) for some p in N ).The space ( S ) is called the space of stochastic test functions and ( S ) ∗ thespace of Hida distributions. Since ( S ) ∗ is the dual space of ( S ) . We will note << , >> the duality bracket between ( S ) ∗ and ( S ) . If φ and Φ both belong to ( L ) then we have the equality << Φ , ϕ>> = < Φ , ϕ > ( L ) = E [Φ ϕ ] .A function Φ : R → ( S ) ∗ is called a stochastic distribution process, or an ( S ) ∗ − process, or a Hida process. A Hida process Φ is said to be differentiableat t ∈ R if lim r → r − (Φ( t + r ) − Φ( t )) exists in ( S ) ∗ .The S -transform of an element Φ of ( S ∗ ) , noted S (Φ) , is defined as thefunction from S ( R ) to R given, for every η in S ( R ) , by S (Φ)( η ) := << Φ , e <.,f> − | f | >> , where ( | | p ) p ∈ Z is the family norms defined by | f | p := P + ∞ k =0 (2 k + 2) p < f, e k > L ( R ) , for all ( p, f ) in Z × L ( R ) .Finally for every (Φ , Ψ) ∈ ( S ) ∗ × ( S ) ∗ , there exists a unique element of ( S ) ∗ ,called the Wick product of Φ and Ψ and noted Φ ⋄ Ψ , such that S (Φ ⋄ Ψ)( η ) = S (Φ)( η ) S (Ψ)( η ) for every η in S ( R ) . We now introduce two operators, denoted M H and ∂M H ∂H , that will proveuseful for the definition of the integral with respect to fBm and mBm. Operators M H and ∂M H ∂H Let H be a fixed real in (0 , . Following [8] and references therein, de-fine the operator M H , specified in the Fourier domain, by ◊ M H ( u )( y ) := √ πc H | y | / − H b u ( y ) for every y in R ∗ . This operator is well defined on the ho-mogeneous Sobolev space of order / − H , denoted L H ( R ) and definedby L H ( R ) := { u ∈ S ′ ( R ) : b u ∈ L loc ( R ) and k u k H < + ∞} , wherethe norm k k H derives from the inner product <, > H defined on L H ( R ) by < u, v > H := c H R R | ξ | − H “ u ( ξ ) “ v ( ξ ) dξ and where c H was given in (1.1).The definition of the operator ∂M H ∂H is quite similar. More precisely, define forevery H in (0 , , the space Γ H ( R ) := { u ∈ S ′ ( R ) : b u ∈ L loc ( R ) and k u k δ H ( R ) < + ∞} , where the norm k k δ H ( R ) derives from the inner product on Γ H ( R ) de-fined by < u, v > δ H := c H R R ( β H + ln | ξ | ) | ξ | − H “ u ( ξ ) “ v ( ξ ) dξ . Following[11], define the operator ∂M H ∂H from (Γ H ( R ) , <, > δ H ( R ) ) to ( L ( R ) , <, > L ( R ) ) ,in the Fourier domain, by: ÿ ∂M H ∂H ( u )( y ) := − ( β H + ln | y | ) √ πc H | y | / − H b u ( y ) ,for every y in R ∗ . The reader interested in the properties of M H and ∂M H ∂H may refer to [11, Sections . and . ]. Fractional and multifractional White noise
Recall the following result ([11, (5.10)]): Almost surely, for every t , B ht = + ∞ X k =0 Ä R t M h ( t ) ( e k )( s ) ds ä < ., e k > . (3.1)We now define the derivative in the sense of ( S ) ∗ of mBm. Define the ( S ∗ ) -valued process W h := ( W ht ) t ∈ [0 , by W ht := + ∞ X k =0 î ddt Ä R t M h ( t ) ( e k )( s ) ds äó < ., e k > . (3.2) Theorem-Definition 3.1 [11, Theorem-definition 5.1] The process W h de-fined by (3 . is an ( S ) ∗ -process which verifies, in ( S ) ∗ , the following equality: W ht = + ∞ X k =0 M h ( t ) ( e k )( t ) < ., e k > + h ′ ( t ) + ∞ X k =0 Ä R t ∂M H ∂H ( e k )( s ) (cid:12)(cid:12)(cid:12) H = h ( t ) ds ä < ., e k > . (3.3) Moreover the process B h is ( S ) ∗ -differentiable on [0 , and verifies dB h dt ( t ) = W ht in ( S ) ∗ . When the function h is constant and identically equal to H , we will write W H := ( W Ht ) t ∈ [0 , and call the ( S ) ∗ -process W H a fractional white noise.Note that (3.3) may be written as W ht = W h ( t ) t + h ′ ( t ) ∂ B ∂H ( t, h ( t )) , (3.4)where W h ( t ) t is nothing but W Ht | H = h ( t ) and where the equality holds in ( S ) ∗ . Since the objects we are dealing with are no longer random variables ingeneral, the Riemann or Lebesgue integrals are not relevant here. However,taking advantage of the fact that we are working with vector linear spaces,we may use Pettis or Bochner integrals. In the frame of the Wick-Itô integral,and in view of the result that will provided by Lemma 3.1 below, the use ofBochner integral appears to be relevant. Indeed, Lemma 3.1 gives an easycriterion for integrability, w.r.t. fBm, of any ( S ) ∗ -valued process Y . Thuswe give here a brief statement on Bochner integral. However, and in ordernot to weigh down this statement we will only give the necessary tools toproceed (see [10, p. ] for more details about Bochner integral). Definition 3.1 (Bochner integral [10], p. ) Let I be a Borel subset of [0 , and Φ := (Φ t ) t ∈ I be an ( S ) ∗ -valued process verifying:(i) the process Φ is weakly measurable on I i.e. the map t << Φ t , ϕ >> is measurable on I , for every ϕ in ( S ) .(ii) there exists p ∈ N such that Φ t ∈ ( S − p ) for almost every t ∈ I and t
7→ k Φ t k − p belongs to L ( I ) . Then there exists an unique element in ( S ) ∗ , noted R I Φ u du , called theBochner integral of Φ on I such that, for all ϕ in ( S ) , << Z I Φ u du, ϕ >> = Z I << Φ u , ϕ >> du. In this latter case one says that Φ is Bochner-integrable on I with index p . Proposition 3.1 If Φ: I → ( S ) ∗ is Bochner-integrable on I with index p then k R I Φ t dt k − p ≤ R I k Φ t k − p dt. Theorem 3.1 ([10], Theorem 13.5)
Let
Φ := (Φ t ) t ∈ [0 , be an ( S ) ∗ -valuedprocess such that:(i) t S (Φ t )( η ) is measurable for every η in S ( R ) .(ii) There exist p in N , b in R + and a function L in L ([0 , , dt ) suchthat, for a.e. t in [0 , , | S (Φ t )( η ) | ≤ L ( t ) e b | η | p , for every η in S ( R ) .Then Φ is Bochner integrable on [0 , and R Φ( s ) ds ∈ ( S − q ) for every q > p such that be D ( q − p ) < where e denotes the base of the natural logarithmand where D ( r ) := r P + ∞ n =1 1 n r for r in (1 / , + ∞ ) . The fractional Wick-Itô integral with respect to fBm (or integral w.r.t. fBmin the white noise sense) was introduced in [8] and extended in [4] using thePettis integral. As we will see in Lemma 3.1 below, the Bochner integrabilityof an ( S ) ∗ -valued process Y is a simple condition that ensures the Wick-Itôintegrability of Y with respect to fBm (see Definition 3.2 below) of any Hurstindex H in (0 , . For this reason, we now particularize the fractional Wick-Itô integral with respect to fBm (or Wick-Itô integral w.r.t. fBm) of [8] and[4] in the framework of the Bochner integral. Definition 3.2 (Wick-Itô integral w.r.t fBm in the Bochner sense)
Let H ∈ (0 , , I be a Borel subset of [0 , , B H := ( B Ht ) t ∈ I be a fractionalBrownian motion of Hurst index H , and Y := ( Y t ) t ∈ I be an ( S ) ∗ -valuedprocess verifying:(i) there exists p ∈ N such that Y t ∈ ( S − p ) for almost every t ∈ I ,(ii) the process t Y t ⋄ W Ht is Bochner integrable on I .Then, Y is said to be Bochner-integrable with respect to fBm on I and itsintegral is defined by: Z I Y s d ⋄ B Hs := Z I Y s ⋄ W Hs ds. (3.5)1 Remark 3.1
In order to keep the name given in [8], we also call this integralfractional Wick-Itô integral.The following lemma ensures us that every Bochner integrable process isintegrable on [0 , w.r.t. fBm of any Hurst index H in (0 , . For sake ofnotational symplicity one denotes, for every integer p , q ( p ) the integerequal to max { p + 1; 3 } if p ≥ and equal to if p = 0 . Lemma 3.1
Let Y := ( Y t ) t ∈ [0 , be an ( S ) ∗ -valued process, Bochner inte-grable of index p ∈ N . Then Y is integrable on [0 , , with respect to fBmof any Hurst index H , in the Bochner sense. Moreover, for any H in (0 , , R [0 , Y s d ⋄ B Hs belongs to ( S − q ( p ) ) . Proof.
Fix H ∈ (0 , and an integer p ≥ .The map t Y t ⋄ W Ht isweakly measurable since t S ( Y t ⋄ W Ht )( η ) is measurable for all η in S ( R ) . Using [10, Remark p. ], one obtains that, for almost all t in [0 , , k Y t ⋄ W Ht k − q ( p ) ≤ k Y t k − p k W Ht k − p . Hence Y t ⋄ W Ht belongs to ( S − q ( p ) ) .Since the map t
7→ k W Ht k − r is continuous for all integer r ≥ (see [11,Proposition 5.9]), one also gets: R k Y t ⋄ W Ht k − q ( p ) dt ≤ ( sup t ∈ [0 , k W Ht k − p ) R k Y t k − p dt < + ∞ . This shows that t Y t ⋄ W Ht is Bochner-integrable of index q ( p ) .Let us now assume that p belongs to { , } . It is sufficient to check thatTheorem 3.1 applies. Condition ( i ) is obviously fulfilled. Moreover, using[10, p. ], we obtain that, for every ( t, η ) in [0 , × S ( R ) , | S ( Y t ⋄ W Ht )( η ) | ≤ k Y t k − p sup t ∈ [0 , k W Ht k − e | η | =: L ( t ) e | η | . Since Y is Bochner integrable of index p , it is clear that L belongs to L ([0 , , dt ) . Moreover, e D ( r − p ) < , for every r ≥ p + 2 . Theorem3.1 then allows to conclude that t Y t ⋄ W Ht is Bochner integrable of index q ( p ) . (cid:3) We end this section with the following lemma, the proof of which is obviousin view of Proposition 2.2, that will be useful in the proof of Theorem 4.1below.
Lemma 3.2
For every p in N , the map ( t, H ) ∂ B ∂H ( t, H ) is continuousfrom [0 , into (( S − p ) , k k p ) . In particular, for every subset [ a, b ] of (0 , ,there exists a positive real κ such that: ∀ p ∈ N , sup ( s,H ) ∈ [0 ,t ] × [ a,b ] k ∂ B ∂H ( s, H ) k − p ≤ κ. (3.6)2 Our aim in this section, is to construct an integral w.r.t. mBm usingapproximating integrals w.r.t. fBms. This new integral, that will be namedlimiting fractional Wick-Itô integral, is defined at the end of this section.The main result of this section is Theorem 4.1, which requires the resultgiven in Lemma 4.1 below.Let p be a fixed integer and Y := ( Y t ) t ∈ [0 , be an ( S − p ) -valued process ( i.e. Y t belongs to for every real t in [0 , and t Y t is measurable from (0 , to ( S − p ) , endowed with its Borelian measure). As explained in Section 2.2,we wish to define the integral w.r.t. an mBm B h , noted R Y t d ⋄ B ht , by aformula of the kind: Z Y t d ⋄ B ht := lim n →∞ Z Y t d ⋄ B h n t + Z h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt, (4.1)where the limit holds in ( S ) ∗ . For this formula to make sense, it is certainlynecessary that Y be Bochner-integrable with respect to fBm, on [0 , , ofall exponents α in h ([0 , . The following technical lemma will be useful toestablish Theorem 4.1 below. Lemma 4.1
For any [ a, b ] ⊂ (0 , and any integer p ≥ , there exists apositive real γ p such that, for all ( t, α, α ′ ) ∈ [0 , × [ a, b ] , k W αt − W α ′ t k − p ≤ γ p | α − α ′ | . Proof.
The interval [ a, b ] in (0 , and an integer p ≥ being fixed, onecan write by definition of the ( S ) ∗ -valued process W , for all ( t, α, α ′ ) in [0 , × [ a, b ] : k W αt − W α ′ t k − p = + ∞ X k =0 ( M α ( e k )( t ) − M α ′ ( e k )( t )) (2 k +2) p . Besdides, for all ( t, k ) in [0 , × N , the function α M α ( e k )( t ) is differ-entiable on (0 , (this is shown in [11, Lemma 5.5]). Using point 1 of [11,Lemma 5.6] and the mean value theorem, one obtains the following fact:there exists a positive real ρ such that for all ( t, α, α ′ , k ) ∈ [0 , × [ a, b ] × N : | M α ( e k )( t ) − M α ′ ( e k )( t ) | ≤ ρ ( k + 1) / ln( k + 1) | α − α ′ | . As a consequence, we get k W αt − W α ′ t k − p ≤ | α − α ′ | ρ P + ∞ k =0 ( k +1) / ln ( k +1)2 p ( k +1) p =: | α − α ′ | γ p . Since p ≥ , γ p is finite and the proof is complete. (cid:3) Y on [0 , is sufficient to guaranteethat both the sequence ( R Y t d ⋄ B h n t ) n ∈ N and the quantity R h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt exist and belong to ( S ) ∗ . It also establishes that the sequence ( R Y t d ⋄ B h n t ) n ∈ N converges in ( S ) ∗ . For any integer p , q ( p ) still denotesan integer defined as before Lemma 3.1 Theorem 4.1
For any process Y := ( Y t ) t ∈ [0 , that is Bochner integrable on [0 , of index p , the sequence ( R Y t d ⋄ B h n t ) n ∈ N and the quantity R h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt are well defined in ( S ) ∗ and both belong to ( S − q ( p ) ) . Moreoverthe sequence ( R Y t d ⋄ B h n t ) n ∈ N converge in ( S − q ( p ) ) to R Y t ⋄ W h ( t ) t dt . Proof.
The existence of the sequence ( R Y t d ⋄ B h n t ) n ∈ N in ( S − q ( p ) ) N isa straightforward consequence of Lemma 3.1 since it has been proven therethat R Y t d ⋄ B Ht is well defined, for any H in (0 , , and belongs to ( S − q ( p ) ) ; q ( p ) being independent from H . The scheme of the proof of the existenceof R Y t ⋄ W h ( t ) t dt is the same that the one we used, in the proof of Lemma3.1, to show the existence of R Y t ⋄ W Ht dt . One only needs to show that sup t ∈ [0 , k W h ( t ) t k − p is finite for any p ≥ . Let then p ≥ be fixed. Thanksto (3.3), (3.4) and to the upper-bound given in [11, Theorem 3.7 point 3],one gets: U p : = sup t ∈ [0 , k W h ( t ) t k − p = sup t ∈ [0 ,
1] + ∞ X k =0 ( M h ( t ) ( e k )( t )) (2 k + 2) − p ≤ ̺ h + ∞ X k =0 (2 k + 2) − p − / , where ̺ h := D sup H ∈ h ([0 , c H ; D being given in [11, Theorem 3.7 point 3] and c H being defined right after Formula (1.1). Since U p is finite as soon as p ≥ ,the existence of R Y t ⋄ W h ( t ) t dt is established.In order to show the existence of R h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt , one just needs toshow that the map t h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) is Bochner integrable on [0 , .Using the same arguments as in the proof of Lemma 3.1 one easily prove that t h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) is weakly measurable on [0 , . Lemma 3.2 entailsthat, sup ( s,H ) ∈ σ h k ∂ B ∂H ( s, H ) k − p ≤ κ for every p , where σ h := [0 , × h ([0 , .We hence get, k h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) k − q ( p ) ≤ k Y s k − p ( sup s ∈ [0 , | h ′ ( s ) | ) sup s ∈ [0 , k ∂ B ∂H ( s, h ( s )) k − p . δ ∈ R ∗ + , such that R k h ′ ( s ) Y s ⋄ ∂ B ∂H ( s, h ( s )) k − q ( p ) ds ≤ δ R k Y s k − p ds < + ∞ . As a consequence, R h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt is welldefined in the sense of Bochner.Finally it just remains to show the convergence, in ( S − q ( p ) ) , of ( R Y t d ⋄ B h n t ) n ∈ N to R Y t ⋄ W h ( t ) t dt . In view of the definition of the functions h n , (2.5)and (3.5), the equality R Y t d ⋄ B h n t = R Y t ⋄ W h n t dt is obvious for every n in N . Setting I n := k R Y t ⋄ W h ( t ) t dt − R Y t d ⋄ B h n t k − q ( p ) and usingProposition 3.1, [10, Remark (2) p.92] and Lemma 4.1, one then has: I n = k Z Y t ⋄ ( W h ( t ) t − W h n ( t ) t ) dt k − q ( p ) ≤ Z k Y t k − p k W h ( t ) t − W h n ( t ) t k − p dt ≤ γ p Z k Y t k − p | h ( t ) − h n ( t ) | dt. The Dominated convergence theorem of Lebesgue finally allows us to writethat lim n → + ∞ I n = 0 and thus achieves the proof. (cid:3) Remark 4.1
The previous proof shows in particular that one does not needthe pointwise convergence of ( h n ) n ∈ N to h on the whole interval [0 , butonly almost everywhere.Define the set Λ p by setting: Λ p := { Y := ( Y t ) t ∈ [0 , ∈ ( S − p ) R : Y is Bochner integrable of index p on [0 , } . Corollary 4.1
Let Y be in Λ p . Then the quantity I hp := lim n →∞ Z Y t d ⋄ B h n t + Z h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt, where the limit and the equality both hold in ( S − q ( p ) ) , is well-defined andbelongs to ( S − q ( p ) ) . Moreover one has the equality: I hp = Z Y t ⋄ W h ( t ) t dt + Z h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt. (4.2)As a consequence of Theorem 4.1 and Corollary 4.1 , the integral w.r.t. mBmexists as a limit of integrals w.r.t. fBms plus a second term. Thus, we arefinally able to define our integral: Definition 4.1 (Limiting fractional Wick-Itô integral)
For any fixed in-teger p and any element Y := ( Y t ) t ∈ [0 , of Λ p , the integral of Y withrespect to B h can be obtained as limits of fractional Wick-Itô integral. Wenote R Y t d ⋄ B ht this integral and call it limiting fractional Wick-Itô integral.It is defined by: Z Y t d ⋄ B ht := I hp = lim n →∞ Z Y t d ⋄ B h n t + Z h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) dt, (4.3)5In view of (4.1) and of Remark 3.1, and even if R Y t d ⋄ B ht is not only alimit of sums of fractional Wick-Itô integrals, the name limiting fractionalWick-Itô integral to call R Y t d ⋄ B ht seems to be indicate since it give theessence of it. Remark 4.2
The linearity of limiting fractional Wick-Itô integral as well asthe equality R ba d ⋄ B ht a.s. = B hb − B ha , for any ( a, b ) in R such that a < b , areconsequences of Definition 4.1.Moreover, for any ( S − p ) -valued process Y := ( Y t ) t ∈ [0 , that admits a lim-iting fractional Wick-Itô integral over a Borel subset I of R , if R I X s d ⋄ B hs belongs to ( L ) ., then E [ R I X s d ⋄ B hs ] = 0 .We shall compare the integral w.r.t. mBm, obtained in Definition 4.1, tothe one defined with the direct approach of [11]. This the goal of the nextsection. A multifractional Wick-Itô integral with respect to mBm was defined in [11].In addition Itô fromulas (in both weak and strong senses) as well as a Tanakaformula were provided. It is interesting to check whether it coincides with theone provided by Definition 4.1. In that view, we need to adapt the definitionof multifractional Wick-Itô integral with respect to mBm given in [11], whichused Pettis integrals, to deal with Bochner integrals.
Definition 5.1 (Multifractional Wick-Itô integral in Bochner sense)
Let I be a Borelian connected subset of [0 , , B h := ( B ht ) t ∈ I be a multifrac-tional Brownian motion and Y := ( Y t ) t ∈ I be a ( S ) ∗ -valued process such that:(i) There exists p ∈ N such that Y t ∈ ( S − p ) for almost every t ∈ I ,(ii) the process t Y t ⋄ W ht is Bochner integrable on I . Y is then said to be integrable on I with respect to mBm in the Bochnersense or to admit a multifractional Wick-Itô integral. This integral, noted R I Y s dB hs ds , is defined by R I Y s dB hs ds := R I Y s ⋄ W hs ds . Remark 5.1
From the definition of ( W ht ) t ∈ [0 , [11, Proposition . ], and theproof of Lemma 3.1, it is clear that every ( S ) ∗ -valued process Y := ( Y t ) t ∈ I which is Bochner integrable on I , of index p , is integrable on I with respectto mBm, in the Bochner sense. Moreover R [0 , Y t dB ht belongs to ( S − q ( p ) ) ,where q ( p ) was defined just before Lemma 3.1.6 In order to compare our two integrals with respect to mBm when they bothexist, it seems natural to assume that Y = ( Y t ) t ∈ [0 , is a Bochner integrableprocess of index p ∈ N . We keep notations of the previous sections, inparticular for p and q ( p ) . Theorem 5.1
Let Y = ( Y t ) t ∈ [0 , be a Bochner integrable process of index p ∈ N . Then Y is integrable with respect to mBm in both senses of Defi-nition 4.1 and Definition 5.1. Moreover R I Y t dB ht dt and R [0 , Y t d ⋄ B ht areequal in ( S ∗ ) . Proof:
The existence of both integrals R [0 , Y t d ⋄ B ht and R I Y t dB ht dt isobvious in view of Theorem 4.1 and Remark 5.1. Moreover Equalities (4.3),(4.2) and (3.4) allow us to write: Z Y t d ⋄ B ht = Z (cid:16) Y t ⋄ W h ( t ) t + h ′ ( t ) Y t ⋄ ∂ B ∂H ( t, h ( t )) (cid:17) dt = Z Y t ⋄ W ht dt. = Z Y t dB ht . (cid:3) Acknowledgments
The author thanks Jacques Lévy Véhel for many helpful remarks and com-ments and the anoymous referee for his suggestions that highly improvedSections 4 and 5 of this paper.
References [1]
E. Alòs and
O. Mazet and
D. Nualart , Stochastic calculus with respect to Gaus-sian processes,
Ann. Probab. , , (2001), 766-801.[2] A. Ayache and
S. Cohen and
J. Lévy Véhel , The covariance structure of mul-tifractional Brownian motion, with application to long range dependence (extendedversion) , in ICASSP, 2000.[3]
A. Benassi and
S. Jaffard and
D. Roux , Elliptic Gaussian random processes,
Rev. Mat. Iberoamericana , , (1997), 19-90.[4] C. Bender , An Itô formula for generalized functionals of a fractional Brownianmotion with arbitrary Hurst parameter,
Sto. Pro. and their App. , , (2003), 81-106.[5] C. Bender , An S -transform approach to integration with respect to a fractionalBrownian motion, Bernoulli , , (2003), 955-983.[6] F. Biagini and
A. Sulem and
B. Øksendal and
N. Wallner , An introductionto white-noise theory and Malliavin calculus for fractional Brownian motion,
Proc.Royal Society, special issue on stochastic analysis and applications , (2004), 347-372.[7]
B. Boufoussi and
M. Dozzi and
R. Marty , Local time and Tanaka formula for aVolterra-type multifractional Gaussian process,
Bernoulli , , (2010), 1294-1311. [8] R.J. Elliott and
J. van der Hoek , A general fractional white noise theory andapplications to finance,
Mathematical Finance , , (2003), 301-330.[9] E. Herbin and
J. Lebovits and
J. Lévy Véhel , Stochastic integration with re-spect to multifractional Brownian motion via tangent fractional Brownian motion,
Preprint , available at http://hal.inria.fr/hal-00653808/fr/ , (2012).[10] H. Kuo , White Noise Distribution Theory , CRC-Press, 1996.[11]
J. Lebovits and
J. Lévy Véhel , White noise-based stochastic calculus with respectto multifractional Brownian motion, to appear in Stochastics , (2012).[12]
R. Peltier and
J. Lévy Véhel , Multifractional Brownian motion: definition andpreliminary results, rapport de recherche de l’INRIA, n .[13] G. Samorodnitsky and
M. Taqqu , Stable Non-Gaussian Random Processes , Chap-mann and Hall/C.R.C, 1994.[14]
S. Stoev and
M. Taqqu , How rich is the class of multifractional Brownian motions?,
Stochastic Processes and their Applications , , (2006), 200-221. Joachim Lebovits