Stochastic differential equations with fractal noise

Transcription

1 Math. Nachr. 78, No. 9, (5) / DOI 1.1/mana.3195 Stochatic differential equation with fractal noie M. Zähle 1 1 Mathematical Intitute, Univerity of Jena, 7737 Jena, Germany Received 8 October, revied 3 July 3, accepted 4 July 3 Publihed online 8 June 5 Key word Forward integral, quadratic variation proce, anticipating tochatic differential equation, Beov pace MSC () 6H1, 6H5, 6H15, 34F5 Stochatic differential equation in R n with random coefficient are conidered where one continuou driving proce admit a generalized quadratic variation proce. The latter and the other driving procee are aumed to poe ample path in the fractional Sobolev pace W β for ome β>1/. The tochatic integral are determined a anticipating forward integral. A pathwie olution procedure i developed which combine the tochatic Itô calculu with fractional calculu via norm etimate of aociated integral operator in W α for <α<1. Linear equation are conidered a a pecial cae. Thi approach lead to fat computer algorithm baing on Picard iteration method. c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction For application in mathematical finance (ee, e.g. [1]) and other field the following tochatic differential equation in R n i relevant: ( ) m ( ) dx(t) = a X(t),t dw (t)+ a j X(t),t db H j (t)+b ( X(t),t ) dt, X(t ) = X. j=1 Here W i the one-dimenional Wiener proce and the B Hj denote fractional Brownian motion with Hurt exponent H j which may vary in time. They may be introduced by mean of the repreentation B Hj (t) := e itu 1 u Hj(t)+1/ dw j (u) for Wiener procee W 1,...,W m. (For the contant H j we obtain claical fractional Brownian motion.) We will conider only the cae H j (t) H> 1 and aume that all H j(t) are Hölder continuou function. In thi cae the ample path of the B Hj have almot urely nice fractional moothne propertie: They are Hölder continuou of all order le than H and they poe finite p variation, p> 1 H, and fractional derivative of all order le than H. Moreover, they are element of the Sobolev Slobodeckij (or Beov) pace W H which are mot appropriate to our approach. Thi guarantee that under moothne aumption on the random vector field a i the tochatic integral in the econd ummand of the above equation for a uitable notion of olution may be undertood in the ene of a.. convergence of the Riemann Stieltje um. The tochatic integral for the firt ummand, i.e. the Brownian motion component, may be determined in the ene of uniform convergence in probability of the Riemann Stieltje um. (We do not aume any adaptedne on the random vector field a,a 1,...,a m,b.) zaehle@math.uni-jena.de c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 198 Zähle: Stochatic differential equation with fractal noie In particular, the correponding linear equation may be ued for modelling tock price development and option pricing in mathematical finance. The fractional Brownian motion component lead to long range dependence and the Wiener proce guarantee trong no-arbitrage (ee [15]). In general, the above equation cannot be treated by mean of emimartingale theory (for adapted coefficient). Becaue of the Brownian motion component it alo doe not fit into the model uing finite p-variation, Hölder condition, or fractional derivative, where the olution depend continuouly on the driving procee. Thee uppoe integrand and integrator of ummed order of fractional moothne greater than 1. In recent more abtract paper the cae of lower order Hurt exponent ha been conidered for different type of tochatic integral, but the olution procedure are complicated. A numerically more acceible approach to the pathwie olution conit in the following. (For implicity it i demontrated here on the time homogeneou cae, the general cae will be treated below.) A in the Do Suman approach for m =we eek the olution pathwie in the form X(t) = h(y (t),w(t)) for ome mooth function h atifying h y (y, z) = a (h(y, z)) and an unknown vector proce Y (t). Then we apply the Itô formula which will be adopted a a general calculation rule in order to determine an auxiliary tochatic differential equation where dw (t) i eliminated: dy (t) = ã i (Y (t),w(t)) db Hi (t)+ b(y (t),w(t)) dt i=1 with new random vector field ã 1,...ã m, b. HereW (t) may be conidered a a parameter function (of le order of moothne than that of the integrator). Such type of equation have been treated in our paper [13] in a more pecial ituation. Extending thi approach to mapping depending on parameterfunction a abovewe get pathwie a unique local olution Y (t) with coordinate in the Beov pace W H. Subtituting thi Y (depending on the choice of h) in the above formula for X(t) we indeed obtain a olution. We will how that the olution i unique in the cla of all procee with generalized quadratic variation atifying the Itô calculation rule. (For different h we obtain different repreentation of the ame proce.) Note that the (random) function h and the auxiliary proce Y maybe determined by Picard iteration method. The eential part of our approach i the above auxiliary SDE for the proce Y. It i olvable for more general driving procee and parameter procee than B H1,...,B Hm and W, rep. Moreover, the method of howing convergence in probability of the Riemann Stieltje um for the firt integral in the above equation by mean of the Taylor expanion i well-known from the literature and goe back to Föllmer [5]. In contrat to other paper we derive thi convergence from that of the remaining integral. The latter can be hown by at leat three different method uing pathwie fractional moothne propertie of integrand and integrator of ummed order greater than 1 (cf. Section.1). The Wiener proce W will be replaced by an arbitrary continuou proce Z with generalized quadratic variation [Z] W H if the firt integral i undertood in a imilar ene. Intead of fractional Brownian motion B H we will chooe arbitrary procee with ample path in W H,whereH>1/. Stochatic integral and procee with generalized quadratic variation.1 Convergence of Riemann Stieltje um and related function pace The following determinitic model may be applied to the ample path of tochatic procee. For real-valued functionf and g on an interval [,T] the extended Lebegue Stieltje integral fdg, t (,t], have been determined in the following three ituation and agree with each other on the common pace of definition: c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3 Math. Nachr. 78, No. 9 (5) / (i) L. C. Young [11] proved (non-abolute uniform in t) convergence of the Riemann Stieltje um in the cae where f and g have finite p-andq-variation Var p f and Var q g with 1 p + 1 q > 1 and do not poe common point of dicontinuity. Moreover, fdg Var p f Var q g + f() (g(t) g()). A tochatic verion wa conidered in Bertoin [1]. (An introduction to thi approach and more recent development may be found in Dudley and Norvaiša [3].) (ii) For mooth function f and g Feyel and de La Pradelle [4] derived the etimate t fdg := fg d cont f α g β t 1+ε + f() (g(t) g()) where α denote the Hölder norm of order α and <ε<α+ β 1. Therefore the integral extend continuouly to Hölder function f and g of ummed order greater than 1. For uch function they proved uniform (in t) convergence of the Riemann Stieltje um to that integral. (iii) In our paper [1] we introduced fdg := ( 1) α D+ α f +(u) Dt 1 α g t (u) du + f(+)(g(t ) g(+)) where f + (u) :=f(u) f(+), g t (u) :=g(u) g(t ) and D β + ϕ ( Dt ϕ β ) denote the left-ided (rep. right-ided) fractional derivative of order β (, 1) of a function ϕ on the interval (, t) in the ene of [9]. The derivative D+ α f + and Dt 1 α g t areaumedtobeelementofl p and L q, rep., where 1/p +1/q =1.(The definition doe not depend on the choice of α and the correponding propertie of an integral are proved.) For the pecial cae of Hölder continuou function of ummed order greater than 1 (cf. (ii)) uniform convergence of the Riemann Stieltje um to our integral i hown with intrinic method of fractional calculu. Thu, in thi cae the integral in (i), (ii) and (iii) agree. It turn our that for the purpoe of the preent paper the more general Sobolev Slobodeckij pace W α (which coincide up to norm equivalence with the Beov pace B,) α are mot appropriate. The norm in thee Banach pace are given by ( T 1/ ( T T f W α := f(t) dt) (f() f(t)) ) 1/ + t α+1 d dt where <α<1. In order to prove contraction propertie of a related integral operator it i appropriate to replace the L -norm of the function itelf by the L -norm. (For α>1/ thi doe not change the pace W α ince thee function are continuou.) Denote f W α f W α, := ( T T := f L + α. f W (f() f(t)) ) 1/ t α+1 d dt, When retricting to ubinterval (, t) we write W α (, t), etc. Further, W β := α<β W α, etc. The Liouville pace I α + (L )(rep. I α t (L )) (ued in the definition of the integral) are given by the norm f I α + (L ) := f L + D+ α f L D+ α f L. (t ) (t ) (t ) c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

4 11 Zähle: Stochatic differential equation with fractal noie Relationhip between thee norm may be found, e.g., in [4] and [13]. In [13] it i alo hown that the integral in.1 (iii) may be repreented in the form 1 fdg = lim ε u ε 1 f() g t ( + u) g t () d du (.1) ε u where convergence hold uniformly in t (,T]. The limit on the right-hand ide exit under more general aumption on f and g and may be conidered a an extenion of our integral. Note that the kernel εu ε 1 act a the δ-function a ε. Therefore the exitence of lim u f() g t ( + u) g t () u d implie that of the above limit. In the context of uniform convergence in probability the latter correpond to a light extenion of the o-called forward integral of Ruo and Valloi [6] for tochatic procee. More detail and relationhip to other integral may be found in [14].. Stochatic forward integral, quadratic variation and Itô formulae Suppoe now that Y i a tochatic càglàd (left continuou with right limit) proce and Z a tochatic càdlàg (right continuou with left limit) proce on [,T]. Then we define (cf. [13]) ( ) 1 YdZ = YdZ := lim ε u ε 1 Y () Z t ( + u) Z t () d du (.) ε u whenever the right-hand ide i determined, where lim tand for uniform convergence in probability and 1 for 1 lim δ δ with probability 1. Then X(t) := + YdZ i càdlàgand X(t) X(t ) =Y (t)(z(t) Z(t )). Moreover, continuityof Z implie that of X. Remark.1 Ruo and Valloi ue the ame tochatic forward integral, but without averaging in the limit procedure. If Z i a emimartingale and Y an adapted càglàd proce then the integral (.1) agree with the uual Itôintegral + YdZ(ee [14]). The quadratic variation proce (or bracket) [Z](t) i a well-known notion in emimartingale theory. Having in mind the tochatic integral (.) we now define the generalized quadratic variation proce (bracket) 1 [Z](t) := lim ε u ε 1 1 ε u (Z t ( + u) Z t ()) d du +(Z(t) Z(t )) (.3) for any càdlàg proce Z uch that convergence hold uniformly (in t (,T]) in probability. The generalized covariation proce [Y,Z] of two uch procee i introduced analogouly. For the pecial cae of emimartingale thee notion agree with the claical one. Many propertie of our extenion may be proved imilarly a in Ruo Valloi [7] where the cae of non-averaged limit i treated. (For thi and the following relationhip ee [13].) Let now Z be a continuou proce with generalized bracket [Z]. Then we get for any random C 1 -function F (z,t) on R [,T] with continuou F and for <t T the imple Itôformula F (Z(t),t) F (Z(),) = (Z(u),u) dz(u)+ x t (Z(u),u) du + 1 F x (Z(),) d[z]() (.4) and the tochatic integral i determined in the ene of (.). Further, by definition the proce X with X(t) = AdZ c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

5 Math. Nachr. 78, No. 9 (5) / for ome càglàd proce A atifie the general Itô formula if F (X(t),t) F (X(),) = (X(u),u)A(u) dz(u)+ x + 1 F x (X(u),u)A(u) d[z](u). (X(u),u) du t Higher-dimenional extenion are traightforward. For our purpoe we need the following pecial verion. Firt note a an important conequence of Definition (.3) that any continuou proce Z admitting a generalized bracket [Z] belong to the pace W 1/ with probability 1. F (y, z) i now an R n -valued random C 1 -function on R m R with continuou F and Y =(Y 1,...,Y m ) i a random proce with coordinate ample path in W β, for ome β>1. Then we obtain F (Y (t),z(t)) F (Y (),Z()) = y i (Y (u),z(u)) dy i (u)+ i=1 + 1 F (Y (u),z(u)) d[z](u). (Y (u),z(u)) dz(u) Here the firt m integral are determined in the ene of Section.1 (iii) or, equivalently, by a.. convergence of the Riemann Stieltje um. The integral w.r.t. Z i given by the tochatic forward integral (.). (The aumption on the Y i implie that thee are procee of vanihing quadratic (co)variation. Similarly, the covariation of Y i and Z are zero.) Note that we may chooe, in particular, Y m (t) =t in order to apply the formula to time dependent SDE. Let now Z := Z be a before and uppoe that Z 1,...,Z m are procee with ample path in W β for ome β> 1. (Recall that uch procee are continuou.) Definition. The vector procex with integral repreentation X(t) = X() + i= admit the (random) Itôcalculuif A i dz i (.5) (.6) [ X i,x k] (t) = A j Ak d [ Z ] exit and for any (random) vector function F (under conideration) of cla C the general Itô formula hold true: F (X(t)) F (X()) n = x (X(u)) j Aj i (u) dz i + 1 j=1 i= n j,k=1 F x j x k (X(u)) Aj (u) Ak (u) d [ Z ] (u). (The integral w.r.t. to Z i are defined in the ene of (.).) Again, the time dependent cae i included taking Z m+1 (t) =t, X n+1 (t) =t, A n+1 i, i m, A n m+1 1. Then the moothne of F w.r.t. to the time argument may be relaxed a in (.6) for y. 3 An integral operator and it contraction property We now turn back to the determinitic cae and conider integrator function g W H alway auming that H> 1. Later on g will be replaced by fractional Brownian motion BH or general tochatic procee with (.7) c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

6 11 Zähle: Stochatic differential equation with fractal noie ample path in W H. We alo conider a parameter function ϕ W 1/,. In the tochatic verion ϕ correpond to the Wiener proce W or, more generally to an arbitrary continuou proce Z with generalized quadratic variation [Z]. For fixed t [,T) we conider the integral operator Af := x + ( ) t a(f,ϕ) dg for ome mooth function a. For <γ 1, t <t<tand C > let W γ, (t,t; x,c) be the et of function on (t,t) uch that f(t +) = x and f t+ W γ, (t,t) C. The following reult provide a local contraction principle for the operator A: Theorem 3.1 Let x,y R, a C 1 (R R, R) and a x (x, y), a y (x, y) be locally Lipchitz in x and g W H for ome 1/ < H 1. Then for any 1/ < β < Hand poitive contant c, C and K there i ome t (t,t) uch that for any ϕ W 1/, (t,t; y,k) the integral operator Af = x + ( ) t a(f,ϕ) dg map W β, (t,t; x,c) into itelf and we have Af Ah W β, (t,t) c f h W β, (t,t) for all f,h W β, (t,t; x,c). P r o o f. In Theorem.3 of [13] we have hown thi under the tronger aumption that ϕ W β, (t,t; y,k) which wa actually not needed. The argument remain valid when uing Theorem. (i) for α := 1 H+ H β intead of Theorem. (ii) from [13] and regarding in the norm etimate that α<α+β 1 = 1 H β < 1. Note that the above integral are determined in the ene of Section.1. We now will replace the function a by a vector field and define the integral coordinatewie. Then we get the following traightforward higherdimenional extenion w.r.t. to the uual norm in Carteian product of normed pace. Theorem 3. (i) The tatement of Theorem 3.1 remain valid if x R n, y R k, g j W H, a j C 1 (R n R k, R n ) with partial derivative being locally Lipchitz in the firt n argument, j =1,...,l, ϕtake value in R k and f and h in R n with coordinate function a before and the integral operator i given by l ( ) Af := x + a j (f,ϕ) dg j. j=1 t In particular, we obtain the local contraction principle for thi integral operator. (ii) If g j (t) =t for ome j then the condition on a j may be relaxed to meaurability, local boundedne and local Lipchitz continuity w.r.t. the firt n argument. Remark 3.3 The modification in the correponding proof of [13] for the cae (ii) are traightforward. 4 Auxiliary differential equation Turning back to our primary aim we now conider on R n R [,T] the R n -valued C 1 -vector field ã j (y, z, t) uch that all partial derivative are locally Lipchitz in the firt n variable y, j =1,...,l, and the locally bounded meaurable vector field b(y, z, t) being Lipchitz in y. Letz (t) be a real-valued parameter function from W 1/, and z 1,...,z l W H,H> 1, be driving function for the differential equation ( dy(t) = ã j y(t),z (t),t ) dz j (t)+ b ( y(t),z (t),t ) dt, j=1 (4.1) y(t ) = y, where t (,T). c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

7 Math. Nachr. 78, No. 9 (5) / Thi equation become exact via integration in the ene of Section.1 (The lat ummand may be included into the um defining z m+1 (t) :=t. Conidering the time argument t in the vector field a an additional parameter function ϕ(t) =t we obtain an integral equation correponding to Theorem 3..) Theorem 4.1 Under the above condition for any 1 <β<hthere exit an interval (t 1,t ) [,T] containing t with the following propertie. Equation (4.1) ha a olution y on (t 1,t ) with coordinate function in W β, (t 1,t ). It may be determined by mean of Picard iteration method which i contractive. A W 1/, - olution i unique on the maximal interval of definition and belong to W H. P r o o f. The firt part i completely analogou to that of Theorem 6.1 in [13] uing Theorem 3. above. The contraction principle provide a unique local W β 1/, -olution. If we have any local W, olution y then Theorem 1. (iii) in [13] applied to the integral repreentation of y yield y W β, on the interval of definition for any β<h. ( Chooe there α := 1 H + H β. ) Thi theorem i an eential tool for olving the above tochatic differential equation. 5 Stochatic differential equation In thi ection our main reult will be preented. Firt we chooe (random) vector field a,a 1,...,a m,batifying the condition (C1) a j C 1 (R n [,T], R n ), all partial derivative are locally Lipchitz in x R n, (C) b C(R n [,T], R n ) i locally Lipchitz in x R n (with probability 1 in the random cae). A in Section., Z i a continuou proce with generalized bracket [Z ] and Z 1,...,Z m are procee with ample path in W H for ome H>1/. X i an arbitrary initial (random) value. Definition 5.1 A local olution X =(X 1,...,X n ) of the SDE dx(t) = a j (X(t),t) dz j (t)+b(x(t),t) dt, j= X(t ) = X i a proce with generalized quadratic variation admitting the (random) Itô calculu with repect to it integral repreentation (5.1) X(t) = X + j= a j (X(),) dz j ()+ b(x(),) d t t in ome neighborhood of t. (Cf. Section.. The tochatic integral are defined by (.). For j =1,...,m thi i equivalent to a.. convergence of the Riemann Stieltje um. Continuity of X i a conequence of the integral repreentation.) In order to determine a pathwie local olution we firt proceed imilarly a in Section 7 of [13]: Conider the auxiliary pathwie differential equation on R n R [,T] h (y, z, t) = a (h(y, z, t),t), h(y,z,t ) = X, where Z :=Z (t ) and Y i an arbitrary random vector in R n. Picard iteration method provide a (non-unique) local olution h C 1 in a neighborhood of (Y,Z,t ) with partial derivative being Lipchitz in y and ( ) h det y (y, z, t). (5.) c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

8 114 Zähle: Stochatic differential equation with fractal noie Moreover, h n (y, z, t) = i=1 We will eek the olution X of (5.1) in the form X(t) = h ( Y (t),z (t),t ) a x i (h(y, z, t),t)ai (h(y, z, t),t). for ome random W H -proce Y to be determined (in dependence of the choice of h and Y (t )=Y ). Applying the Itô formula (6) to the function h we obtain dx(t) = h ( Y (t),z (t),t ) dz (t)+ k=1 n k=1 h y k ( Y (t),z (t),t ) dy k (t) + h ( Y (t),z (t),t ) dt + 1 h( Y (t),z t (t),t ) d [ Z ] (t) n = a (X(t),t) dz h ( (t)+ Y (t),z y k (t),t ) dy k (t)+ h ( Y (t),z (t),t ) dt t + 1 n i=1 Comparing thi with (5.1) we are led to a econd auxiliary SDE: k=1 a ( ( h Y (t),z x i (t),t ),t ) a i ( ( h Y (t),z (t),t ),t ) d [ Z ] (t). n h ( Y (t),z y k (t),t ) dy k (t) ( ( = a j h Y (t),z (t),t ),t ) ( dz j (t)+ b ( h ( Y (t),z (t),t ),t ) h ( Y (t),z (t),t )) dt t j=1 1 n i=1 a ( ( h Y (t),z x i (t),t ),t ) a i ( ( h Y (t),z (t),t ),t ) d [ Z ] (t). In a neighborhood of t it i equivalent to following matrix repreentation. dy (t) = Y (t ) = Y. ( h ( Y (t),z (t),t )) 1 y [ m ( ( a j h Y (t),z (t),t ),t ) dz j (t) j=1 ( + b ( h ( Y (t),z (t),t ),t ) h ( Y (t),z (t),t )) dt t 1 a ( ( h Y (t),z (t),t ),t ) ( ( a h Y (t),z (t),t ),t ) d [ Z ] (t) x ], (5.3) In order to apply Theorem 4.1 we now additionally aume [Z ] W H with the ame H a for Z 1,...,Z m and conclude the exitence of a pathwie local olution Y W H of (5.3) which i unique in W 1/,. It turn out that thi procedure provide the unique olution of our tochatic differential equation: Theorem 5. Suppoe that the random vector field atify (C1) and (C), Z i a continuou proce with generalized bracket [Z ] and [Z ],Z 1,...,Z m are procee with ample path in W H for ome H>1/. c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

9 Math. Nachr. 78, No. 9 (5) / Then any repreentation X(t) = h ( Y (t),z (t),t ) with h atifying (5.) and Y W H locally determined by (5.3) provide a pathwie local olution of the SDE (5.1). If X i an arbitrary olution in the ene of Definition 5.1 then it agree with any of the above repreentation on the common interval of definition. P r o o f. Exploiting Theorem 4.1 we can ue completely the ame argument a in the proof of Theorem in [13]. For brevity we recall only the main idea. Firt note that applying the Itô formula to the function h and uing (5.) and (5.3) one how that h ( X(t),Z (t),t ) i indeed a olution. In order to prove uniquene take any olution X(t) a required and compare it with one of the above h ( Y (t),z (t),t ) : The mapping (y, z, t) (h(y, z, t),z,t) i invertible in a neighborhood of (Y,Z,t ) with differentiable invere mapping (x, z, t) (u(x, z, t),z,t),i.e., u(h(y, z, t),z,t) = y. Then the partial derivative of u and h are in certain relationhip. Thee and the Itôformula(.7)appliedtothe random function u and the proce ( X(t),Z (t) ) given in integral repreentation X(t) = X + j= t a j (X(),) dz j ()+ t b(x(),) d, Z (t) = Z + dz () t lead to an integral repreentation of u ( X(t),Z (t),t ). Subtituting h (y, z, t) =a(h(y, z, t),t) one conclude that the proce Ỹ (t) := u ( X(t),Z (t),t ) ha ample path in W 1/ 1/, and atifie the integral equation (5.3). Since the olution of (5.3) i unique in W, we infer Ỹ (t) =Y (t), hence u( X(t),Z (t),t ) = Y (t) and thu X(t) =h ( Y (t),z (t),t ) in a neighborhood of t. Remark For the cae m =the above model i treated in [13].. In the general cae the quetion of extending the olution to the whole time interval i reduced to that for the auxiliary differential equation (5.) and (5.3), i.e. to growth condition on the random vector field. Ruo and Valloi [8] conidered the cae n =1and dx(t) = σ(x(t)) dz (t)+b(t, X(t)) dv (t) for a proce V of bounded variation in full detail. They extended the Do approach taking for h in (5.) the flow given by h (y, z) = σ(h(y, z)), h(y, ) = y. 3. For n =1and random number σ,σ 1,...,σ m,βour method implie that the unique olution of the linear equation dx(t) = σ i X(t) dz i (t)+βx(t) dt, i= X() = X c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

10 116 Zähle: Stochatic differential equation with fractal noie i given by { m X(t) = X exp σ i Z i (t) 1 [ σ Z ] } (t)+βt i= and imilarly for time dependent coefficient. Note that in the latter cae the C 1 -property of the σ i i not really needed. It may be relaxed, e.g., to the condition that the ummed order of differentiability of σ i and Z i i not le than The cae of Gauian driving procee Z,...,Z m i conidered in Coutin and Decreuefond [] and other paper with method of Malliavin calculu. The notion of Skorohod-type tochatic integral ued there i different from the above verion. Reference [1] J. Bertoin, Sur une intégrale pour le proceu à α variation bornée, Ann. Probab. 17, (1989). [] L. Coutin and L. Decreuefond, Abtract non linear filtering theory in the preence of fractional Brownian motion, Ann. Appl. Probab. 9, (). [3] R. M. Dudley and R. Norvaiša, An introduction to p-variation and Young integral, Techn. Report 1 (MaPhySto, Univerity of Aarhu, 1998). [4] D. Feyel and A. de La Pradelle, Fractional integral and Brownian procee, Potential Anal. 1, (1989). [5] H. Föllmer, Calcul d Itô an Probabilité, Séminaire de Probabilité XV, Lecture Note in Mathematic Vol. 85 (Springer-Verlag, Berlin Heidelberg New York, 1989), pp [6] F. Ruo and P. Valloi, Forward, backward and ymmetric tochatic integration, Probab. Theory Related Field 97, (1993). [7] F. Ruo and P. Valloi, The generalized covariation proce and Itô formula, Stoch. Proce. Appl. 59, (1995). [8] F. Ruo and P. Valloi, Stochatic calculu with repect to continuou finite quadratic variation procee, Stoch. Stoch. Rep. 7, 1 4 (). [9] S. G. Samko, A. A. Kilba, and O. I. Marichev, Fractional Integral and Derivative. Theory and Application (Gordon and Breach, 1993). [1] A. N. Shiryaev, Eential of Stochatic Finance (World Scientific 1998). [11] L. C. Young, An inequality of Hölder type, connected with Stieltje integration, Acta Math. 67, 51 8 (1936). [1] M. Zähle, Integration with repect to fractal function and tochatic calculu I, Probab. Theory Related Field 111, (1998). [13] M. Zähle, Integration with repect to fractal function and tochatic calculu II, Math. Nachr. 5, (1). [14] M. Zähle, Forward integral and tochatic differential equation, in: Seminar on Stochatic Analyi, Random Field and Application, edited by R. C. Dalang, M. Dozzi, and F. Ruo, Progre in Probability Vol. 5 (Birkhäuer Verlag, ), pp [15] M. Zähle, Long range dependence, no arbitrage and the Black Schole formula, Stoch. Dyn., 65 8 (). c 5 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim