Itô Formula For The Two-parameter Fractional Brownian .
Ito formula for the two-parameter fractional Brownian motion1Itô formula for the two-parameter fractional Brownian motionusing the extended divergence operatorCiprian A. Tudor11Frederi G. Viens2SAMOS-MATISSE, Université de Paris 1 Panthéon-Sorbonne ,90, rue de Tolbiac, 75634, Paris, France.tudor@univ-paris1.fr2Dept. Statistics and Dept. Mathematics, Purdue University,150 N. University St., West Lafayette, IN 47907-2067, USA.viens@purdue.edu 1 (765) 494 6035, fax 1 (765) 494 0558July 17, 2005AbstractWe develop a stochastic calculus of divergence type with respect to the fractional Brownian sheet(fBs) with any Hurst parameters in (0; 1), and with respect to two-parameter random elds beyondthe fractional scale. We de ne stochastic integration in the extended Skorohod sense, and derive Itôand Tanaka formulas. In the case of Gaussian elds that are more irregular than fBs for any Hurstparameters, we are able to complete the same program for those Gaussian elds that are almost-surelyuniformly continuous.Key words and phrases: fractional Brownian motion, Brownian sheet, Malliavin calculus, Skorohodintegral, Hurst parameter, Gaussian regularity.AMS 2000 MSC codes: primary 60H07; secondary 60G15, 60G60, 60H051IntroductionIn recent years stochastic integration with respect to Gaussian processes in general and to the fractionalBrownian motion (fBm for short) in particular has been studied intensively. Di erent approaches have beenconsidered in order to develop a stochastic calculus for fBm, including the Skorohod integration based on aduality relation from the Malliavin calculus (see [1] and [2]), the pathwise stochastic calculus –especially the
Ito formula for the two-parameter fractional Brownian motion2regularization integrals of Russo and Vallois (see these authors’original article in [15], or the presentationin [13]), and the rough path analysis (see [10]). In all cases, the situation when the fBm’s so-called Hurstparameter H is small proves to be most di cult. This H is a self-similarity parameter, and is related to theregularity of the fBm in the sense that almost every path of fBm is -Hölder continuous for anynot for H, but H.For example, in the classical Malliavin calculus approach, the integral of fBm with respect to itself existsif and only if H 1 4. In the pathwise approach, the barrier for the standard de nition of the symmetricintegral (generalization of the Stratonovich integral) is H 1 6. Clearly, for small parameter, one needsan extended, relaxed way to integrate. It was recently discovered in [4] that the barrier of H 1 4 canbe overcome by considering a weaker form of Skorohod integration. In [8], a special form of the symmetricintegral was de ned for all H 0 as well. In [11], Skorohod integration and stochastic calculus were extendedfar beyond the scale of H 0, using ideas from [4] and a new, streamlined method whose success is basedon the avoidance of all references to the cumbersome so-called fractional derivatives and integrals.In this article, we choose to use the techniques developped in [11] to study the problem of stochasticcalculus for two-parameter Gaussian processes. The canonical example of such processes is the the fractionalBrownian sheet. It was studied in [16] for H 1 2, where Itô and Tanaka formulas were established, theformer formula being the canonical chain rule of stochastic calculus, its cornerstone, and the latter being arepresentation of fBm’s local time (occupation time density) using a stochastic integral (see [4] and [11] forone-parameter results). The purpose of this article is to show that the techniques of [16], which only applyto the case of H 1 2, can be supplanted by developing a stochastic integration that works also for thefractional Brownian sheet with any Hurst parameters less than 1 2, and beyond the fractional scale, usingthe ideas of [11]. Our generalization of the one-parameter results of [4] and [11] is non-trivial because of theappearance of not one but four Skorohod stochastic integrals in the Itô formula, including two separate newtypes of integrals with respect to two distinct 2-parameter processes. In a one-dimensional situation, an Itôformula can be established by a simple identi cation procedure, where the single Skorohod stochastic integralis proven to exist and is calculated all in one step (a method used in [4] and in [11]). In our situation, theexistence of each Skorohod stochastic integral in the Itô formula has to be dealt with separately beforehand.Then, and only then, can an identi cation procedure be used to establish the Itô formula.The paper is organized as follows. Section 2 contains preliminaries on the standard and extended Malliavincalculus with respect to the fractional Brownian sheet, including a proof of existence of extended Skorohodintegrals. In Section 3 we derive an Itô’s formula for Hurst parameters below 1 2, and brie‡y discuss the localtime, including a Tanaka formula. Section 4 describes the extension of our calculus beyond the fractionalscale, generalizing the approach of [11] to two parameters, including Itô and Tanaka formulas.
Ito formula for the two-parameter fractional Brownian motion22.13PreliminariesMalliavin calculus and Wiener integralLet T [0; 1]2 and let Ws;t;(s;t)2Tbe a fractional Brownian sheet with Hurst parameters;2 (0; 1).This process is de ned as a centered Gaussian process under some probability space ( ; F; P), starting fromzero, and with the covariance functionE Ws;t; Wu;v; R;(s; t; u; v) : 1 2s u221 2t v22uj2jsjtvj2:We can see that for each xed s 2 [0; 1]; Ws; ; is, up to the constant factor s2 , a one-parameter fBm withHurst parameter ; and similarly for W ;t; when t is xed; note that R, as a tensor-product of covariancefunctions (of fBm’s), is itself a bona de covariance function Let us brie‡y recall the framework of the Malliavincalculus for the fractional Brownian sheet. Denote by H(2) the canonical Hilbert space of WH(2) is the closure of the linear space of linear combinations of indicator functions 1[0;s][0;t];. That is,, s; t 2 [0; 1]with respect to the scalar product de ned byh1[0;s][0;t] ; 1[0;u] [0;v] iH(2) R;(s; t; u; v):Let SH(2) be the class of ‘smooth’random variables of the formF f W;('1 ); : : : ; W;'i 2 H(2)('n )where f and all its derivatives are bounded. The Malliavin derivative operator acts on random variables Fas above in the following waynX@fDs;t F W@xii 0;('1 ); : : : ; W;('n ) 'i (s; t)(s; t) 2 T:The operator D is closable and it can be extended to the closure of SH(2) with respect to the normkF k1;2 EkF k2L2 () EkDF k2L2 (;H(2) ) :The classical Skorohod integral is the adjoint of D. Its domain Dom( ) is the class of square integrableprocesses U such that for some constant C 0,jEhDF; U iH(2) jCkF kL2 ()8F 2 SH(2) :One of the long-standing di culties with the Skorohod integral is that, not only is its domain di cult tocharacterize, even in the standard Brownian case (H 1 2), but for low H, its domain is too small to beof any practical purpose. For example, in the one-dimensional case, the fractional Brownian motion B H isintegrable with respect to itself if and only if H 1 4 (see [4]). The same happens in the case of the sheet:
Ito formula for the two-parameter fractional Brownian motionthe argument of [4] can be used to show that W;42 Dom( ) if and only if both 1 4 and 1 4.Therefore, an extended divergence is needed for the stochastic integration with respect to the fractionalBrownian sheet with small Hurst parameters.Before doing so in the next section by following the method of [11] – itself based on ideas of [4] –, weintroduce an operator based on the Kernel representation of fBm. For fBm strictly speaking, it is traditionalto use elements of fractional calculus. Even though part of the point of our work is to avoid the use of suchcalculus, we include the standard formulas here, so that the reader who is used to these objects will be ableto see how it ties into our development. Let f be a function on [0; 1] andZ b1f (s)Ib f (t) : ds( ) t (s t)1 0. Thenis the right-sided Riemann-Liouville fractional integral of order , while the right-sided integral Ia is de nedusing integration from a to t. For2 (0; 1)Db f (t) : 1(1) tZbtf (s)ds:(s t)is the left-sided Riemann–Liouville fractional derivative of order ; IbThen de ne the operator1K ?;t f (s) : c s 2where c 2 (12)1 22 ) B((12; 212It2 ): We seth?K ?;2; ;t;s : K ;t()12and Dbare inverses of each other.if ( ) (s)K ?;s :(1)(2)In the sequel we will simply write K ?;2 by omitting the parameters, if this does not lead to confusion. In thesame way that the operator K ? ;t is the kernel of the well-known Brownian representation of fBm integration,1?;2our operator K ?;2L2 [0; T ] ,; satis es, for any test function f 2 KZ sZ tZ s Z t hi;W(dq; dr) f (q; r) K ?;2; ;t;s f (q; r) dWq dWr ;00q 0r 0where the integrals on either side are of Wiener type.The operator K ?;2; can also be de ned by (2) by using the Kernel representation of fBm, following [1].There is a deterministic function K (t; s) de ned for 0 s t 1 such thatZ sZ tW ; (s; t) K (t; q) K (s; r) dWq dWr :00For completeness, we give the relevant formulas here: for c a constant depending only on ,Z z 1311F (z) cr 2 1 (1 r) 2 dr;2011tK (t; s) c (t s) 2 s 2 F;s3s 12@K1(t; s) c ()(t s) 2:@t2t
Ito formula for the two-parameter fractional Brownian motionWe then haveK ? ;t f (s) K (t; s)f (s) ZsMoreover, ift5@K(r; s) (f (r)@sf (s)) dr: 1 2, we can see that for some constant c0 ,jK (t; s)j@K(r; s)@sc (ts)1 2;(3)c (ts)3 2:(4)In fact, all the developments below, which are based on fBm, can be repeated as based upon any GaussianRtprocess X which can be represented as X (t) 0 K (t; r) dW (r) where K is a non-random function satisfyingthe two relations (3) and (4) above, although the nal form of the Itô and Tanaka formulas may lookunfamiliar. We choose to work speci cally with fBm for the sake of readability. On the other hand thispaper’s last Section 5 deals with a general class of Gaussian processes which is even allowed to go beyondthose that satisfy (3) and (4). The interested reader will easily be able to check that the Itô and Tanakaformulas for a general K follow immediately from the corresponding results in Section 5.2.2Extended integralThe extended integral can be de ned as in the one-dimensional situation in [4] or [11]. We follow thetechniques in [11]: we introduce the Hilbert spaceH(2);0 K ?;2;adj K ?;21L2 (0; 1)where K ?;2;adj is the adjoint of the operator K ?;2 , and we construct the Malliavin derivative D as above,relative to the new space H(2);0 instead of H(2) . Since H(2);0 is smaller than H(2) , this de nition is immediate.We will say that a square integrable process U belongs to the extended domain of the divergence operator,and we write U 2 Dom? ( ), if there exists a random variable (that we shall also denote by (U )) such thatZ ZE(F (U )) E Us;t K ?;2;adj K ?;2 D ; F (s; t) dsdt 8F 2 SH(2);0 :(5)TThis way, shifting the adjoint back onto F , we see that the ‘new’ extended integral restricted to Dom( )coincides with the standard Skorohod integral. In the sequel we will simply write H; H0 instead of H(2) ; H(2);0 .The reader may consult [11] for a proof that H0 is not restricted to constant random variables. In fact, [11]established that H0 is rich enough to guarantee that the above de nition ofvariable in L2 ( ; F; P) if F is the sigma- eld generated by W;(U ) de nes a unique random. This uniqueness is usually called thedetermining class property of SH0 for . It is remarkable to note that, now that our operator K ?;2 isde ned, and the existence and determining class properties of SH0 are established, there will no longer beany reference to the actual form of K ?;2 . We contend that Skorohod integration, extended or not, should notrequire the use of fractional calculus: one should only have to specify how the kernel K ? is de ned, by any
Ito formula for the two-parameter fractional Brownian motion6analytic method, which may or may not refer to factional calculus, and show the space SH0 of test randomvariables F is a determining class. That such characteristics are su cient for developing a full stochasticcalculus is the underlying argument in [11], which proves it in the single-parameter case for a wide class ofVolterra-type processes which span the fractional Brownian scale and go beyond. The present article showsthat the same program can be achieved for two-parameter processes.It is also possible to characterize the extended domain Dom? ( ) using multiple stochastic integrals. Werecall Theorem 3.2 of [9] (proved in the one-parameter case; but it can be immediately extended to thetwo-parameter case). Let u be a square integrable process having the chaos representationXu (s; t) In (fn ( ; (s; t)))n 0where In denotes the multiple integral of order n with respect to W;and fn 2 HnL2 (T ) is symmetric inthe rst n pairs of variables. Then u 2 Dom? ( ) if and only if f n (the symmetrization of fn in all variables)belongs to Hn 1andIn this case (u) 2.3Xn 0Pn 0 In 1 (fn ).(n 1)!jf n j2Hn 1 1:(6)Speci c double integralWe need to introduce a double Skorohod integral that appears in the expression of the Itô formula for thefractional Brownian sheet. To motivate this de nition, let us brie‡y recall some elements of the regular casewhen the parametersformula for (W;and)2(Ws;t; )2 2Zs0Z s;t L2 ( )M(2)lim(2)j j!0In this situation we have the following decompositiont s;t s2 t2Wu;v; dWu;v; 2MmX1 nX1@1[si ;si 1 ][0;tj ] (respect to W;(2))1[0;si ]i 0 j 0denotes the double Skorohod integral with respect to Wclear, let us point out that(7)0 s;t is de ned as the limitwhere the process M0where12.are bigger than;[tj ;tj 1 ] (1)A(8)and the above limit exists. To becan be interpreted as a two-fold iteration of the Skorohod integralwith(e.g. de ned in the previous section): it is not to be confused with the iteration of the singleSkorohod integral with respect to the one parameter process Ws ;in particular, it has four scalar parameterswhich are grouped pairwise. Moreover, the Itô formula for the fractional sheet contains a “speci c” sheetRsRt u;v which is de ned asintegral 0 0 f 00 (Wu;v; )dM01Z sZ tmX1 nX1 u;v L2 ( )f 00 (W ; )u;v dMlim (2) @f 00 Wsi;;tj 1[si ;si 1 ] [0;tj ] ( )1[0;si ] [tj ;tj 1 ] ( )A :00j j!0i 0 j 0(9)
Ito formula for the two-parameter fractional Brownian motion7mndenotes the partition of [0; 1]2 de ned by the two increasing sequences fsi gi 1 and ftj gj 1 , and j jwhereis its mesh. The fact that the parametersandare supposed to be bigger than 1 2 plays an essentialrole in the proof of the convergence of the sequences from the right side of (8) and (9). Therefore, for should be understood in the extended way. Nevertheless, the intuitivesmall parameters, the integral dM is thatinterpretation of M s;t ds Ws;t dt Ws;tdMwhere for example ds Ws;t denotes the di erential of the fBm s 7! Ws;t when t is xed. Accordingly we canformally writeZ0 1ZZ1 u;vg Wu;v; dM0u0 1u0 0 Zv 1v 0u 1u 0ZZv 0 1v 0 0and this shows the the integralto W;RRZu 1uu0 0v 0 1gv 0 0u 0ZZZ1v v 0Wu;v;01[0;u] (u )1[0;v] (v0)dWu;v; 0!dWu0;;vg Wu;v; dWu0;;v dWu;v; 0 can be interpreted as a two-fold iterated integral with respectg(W )dM above entirely rigorous, we. To make the de nition of the stochastic integral with respect to Mnow only need to de ne the double integral as an extended divergence integral.De nition 1 Let U 2 L2 (T(U 2 Dom?(2)T). We say that the process U belongs to the extended domain of) if there exists a random variable(2)(2)(U ) 2 L2 ( ) such that , for every smooth randomvariable F 2 SH(2);0 it holds thatE FZ Z(2)(U ) Z Z(u;v)2T(u0 ;v 0 )2TE U(u;v);(u0 ;v0 ) K ?;2;adj K ?2 D( ; ) K ?;2;adj K ?;2 D(; )F(u; v) (u0 ; v 0 ) dudvdu0 dv 0 :(10)We will also write(2)(U ) : Z ZTZ ZTU(u;v);(u0 ;v0 ) dW;(u; v) dWu0;;v0 :Remark 1 Since the action of the operator L K ?;2;adj K ?2 is deterministic, we have for every smoothrandom variable F ,K ?;2;adj K ?2 D( ; ) K ?;2;adj K ?2 D(; )F(u; v) (u0 ; v 0 ) (L(2)L)(D( ; );(; 0 ) F ) ((u0; v 0 ) ; (u; v))where D(2) is the second iterated Malliavin derivative. Therefore, relation (10) can be written asZhi(2)(2)E F(U ) E U(u;v);(u0 ;v0 ) (L L) D( ; );( ; 0 ) F (u0 ; v 0 ); (u; v) dudvdu0 dv 0 :T2
Ito formula for the two-parameter fractional Brownian motion38Main resultIn this section we derive the Itô formula for the fractional Brownian sheet for any Hurst parameter, by usingthe technique introduced in [4] and [11] based on the extended Skorohod integral. However, there is onecomplication in our situation which was not present in the one-parameter settings of [4] and [11]. In thesetwo works, the Itô formula can be considered as an equality between two terms: an extended Skorohodintegral I, and the sum S of a Riemann integral and a deterministic function of the underlying process. Theidea is then only to show that S I by proving that S satis es the de nition of I in the extended Skorohodsense; indeed, we then obtain the existence of the Skorohod integral and the Itô formula simultaneously. Inour situation, we cannot proceed this way directly because we will have in our Itô formula not one but fourSkorohod integrals with respect to di erent di erentials. Therefore, as a preliminary step, we must showthat three of the four extended Skorohod integrals exist a-priori, so that we may use their de nition to provethe nal result. Throughout, we use the generic notation t for a pair (s; t) 2 T [0; 1]2 . For convenience’snsake, for any function h on T , we also use the abusive notation h(t)1[0;t]( ) for the function de ned on T n 1by the map(t; t1 ; t2 ;n; tn ) 7! h(t)1[0;t](t1 ; t2 ;; tn ):(11)We start with the following result.Lemma 1 Let f 2 C 1 (R) satisfying (13) and put h(t) E f Wtnh(t)1[0;t]( ) 2 H(2): Then it holds that(n 1)for every nand there exist an integer N large enough such that if nn 2kh(t)1[0;t]k (2)(H )Proof:;1:N we have(n 1)C:n(12)Let us prove rst the result when f is a polynomial function; moreover, without loss ofgenerality, let f (x) xp , where p is an even integer (for odd integers, h is null). Then with t (t; s), wehave h(t) cp t p s p . Since H(2) is the tensor product Hilbert space H(2) HH (where H : H isthe canonical space of the one-parameter fBm B with Hurst parameter ), it su ces to prove, using theone-parameter version of the abusive notation (11), thatt p 1[0;t] ( ) 2 H(n 1)nor, equivalently, K ?;n 1 t p 1[0;t](t1 ; : : : ; tn ) 2 L2 ([0; 1]n 1 ) (where K ?;n is the n-fold tensor product oper-ator of K ?;1 , and we use the abusive notation of naming a function by its value). Using the de nition of the
Ito formula for the two-parameter fractional Brownian motion9operator K ?;n 1 it is not di cult to observe thatnkK ?;n 1 t p 1[0;t](t1 ; : : : ; tn k2L2 ([0;1]n 1 )ihn kK ?;1 t p kK ?;n 1[0;t](t1 ; : : : ; tn )k2L2 ([0;1]n ) k2L2 ([0;1])hi2 kK ?;1 t p kK ?;1 1[0;t] ( )k2n2L ([0;1]) kL2 ([0;1])Note rst thatkK ?;1 1[0;t] ( )k2L2 ([0;1]) E Bt2 t2 :Consequently, we only need to prove that the function t(p 2n)has a nite norm in H. To argue this, letus refer to Proposition 7 in [4] which states that if a process u is in Dom? ( ) such that E[u:] 2 L2 (R), thenE[u:] is in H. But t(p 2np) is equal to E B n 2 (t) which belongs to Dom? ( ) due to Lemma 9 in [4]. Theinequality (12) can be proved using e.g. the fact that for xed , there exists N large enough such that thefunction tof K ?;1 t(N p 2)(N p 2)is Lipschitz. Then it can be seen by a straightforward calculation that the L2 [0; 1]-norm(thus the H-norm of t(N 2p)) is bounded by C N , and that this bound is uniform in p.The reader may also refer to the calculations in Section 5, which are valid in all cases including the fractionalBrownian scale, for a proof of estimates such as (12).The general case when f is C 1 follows by a density argument. Let us only point out the main idea. NowZx21h(t) pe 2 f (xt s )dx:2 RThe key point of the proof is to show that the function f (xt )t2nis in H and this can be seen, for example,by using a polynomial approximation of f , the de nition of the operator K ?;1 and the dominated convergencetheorem. Condition (13) assures the existence of the integral with respect to dx.We may now prove our preliminary existence result.Proposition 1 There exists a 0 depending only onandsuch that for any f 2 C 1 (R) such that fand all its derivatives satisfy the conditionjf (x)jM exp (ax2 )for jxj large enough, where M is a positive constant, we have1[0;s0 ];[0;t0 ] (s; t)f (Ws;t) 2 Dom? ( ) for every s0 ; t0 2 [0; 1]:(13)
Ito formula for the two-parameter fractional Brownian motionProof:10We will assume that s0 t0 1; the general case is analogous. Using Stroock’s formula (see[12]) we get,f (Ws;t; ) X 1 hIn D(n) f (n) (Wtn!;)n 0X 1E f (n) (Wtn!n 0X In (gn ( ; t)); in) In 1[0;t]()n 0wheregn ( ; t) 1E f (n) (Wtn!;n) 1[0;t]( ):Here ’’represents n variables. Let us denote by g n the symmetrization of g in n 1 variables. We need toshow that(n 1)g n 2 H(2)andXn 0(14)(n 1)!kg n k2 (2)(H ) 1:(n 1)(15)First, observe that (14) holds due to Lemma 1. Also, we have thatn 1X1g n (t1 ; : : : ; tn 1 ) h(ti )1[0;ti ] (t i )(n 1)! i 0where t i is the vector (t1 ; : : : ; tn 1 ) with ti missing and h(t) is the function h(t) E f (n) (Wt;) . Tocheck (15), we can write using the Lemma 1, that for some N large enoughXn N(n 1)!kg n k2 (2)(H )(n 1) Xn NXn Nn 1X1kh(ti )1[0;ti ] (t i )k2 (2)(H )(n 1)! i 0n 12n 1 Xkh(ti )1[0;ti ] (t i )k2 (2)(H )(n 1)! i 0(n 1)(n 1)X C2n 1 1:nn!n NDenote by Hn the n -th Hermite polynomialH0 (x) : 1 and Hn (x) ( 1)n x2 de2en!dxx22:and recall the basic propertiesDHn (W;(')) Hn1 (W;('))'(16)
Ito formula for the two-parameter fractional Brownian motionfor every ' 2 H(2);0 , andHk1 (W;11('))' kHk (W;('))(17)We state our main result.Theorem 1 Let f 2 C 1 (R) such that f and all its derivatives satisfy (13). Then f 0 (W?Dom ( ), and f00(Wu;v;00?)1[0;s] (u)1[0;t] (v)1[0;u] (u )1[0;v] (v ) 2 Dom ((2)[0;t]21dvdu Wu;v; du0where, by de nition,Z sZ tZZ00; f (Wu;v )dMu;v 0)1[0;s]) and we have the following Itô for-mula for the fractional Brownian sheet:Z sZ t;f (Ws;t ) f (0) f 0 (Wu;v; )dWu;v;00Z sZ tZ sZ t00;21 21 u;v 2f (Wu;v )uvdvdu f 00 (Wu;v; )dM0000Z sZ tZ sZ t000;21 2; f (Wu;v )uv dv Wu;v du f 000 (Wu;v; )u2 v 20000Z sZ tf iv (Wu;v; )u4 1 v 4 1 dvdu 0;0TZZTf 00 (Wu;v; )1[0;s] (u)1[0;t] (v)1[0;u] (u0 )1[0;v] (v 0 )dWu0;;v dWu;v; 0(18)and we recall that du Wu;v; denotes the Skorohod di erential of the one-parameter fractional Brownian motionu ! Wu;v; .Proof: By Proposition 1 it holds that f 0 (W;)1[0;s][0;t]2 Dom? ( ) for every s; t. Similar argumentsallow to show the integrability of the integrand for the other two Skorohod integrals in the right side . The existence of the stochastic integral with respect to M in the Itôexcepting the one involving M , from the second statement in the theorem. This second statement, onformula follows, by de nition of Mmembership in Dom?(2), is not, strictly speaking, contained in Proposition 1, but its proof is a trivialgeneralization to double integrals of the proof of Proposition 1. We omit all details. The existence of theremaining two stochastic integrals in the Itô formula follows trivially from existence results in [4], sincethese stochastic integrals are, by Fubini, with respect to one-parameter fBm’s. Now using the de nitionof the extended divergence integral, it su ces to show, invoking only simple random variables of the form
Ito formula for the two-parameter fractional Brownian motion12F Hn W ; (') ; since they are dense in L2 ( ), thatZ ZZ1[0;s] (u)1[0;t] (v)1[0;u] (u0 )1[0;v] (v 0 )(u;v 0 )2TE Hn2 (Wf (Ws;t; EZs0E(u0 ;v)2TZZZ0s0('))f 00 (Wu;v; ) K ?;2;adj K ?;2 ' (u; v 0 ) K ?;2;adj K ?;2 ' (u0 ; v)dudu0 dvdv 0Z sZ tf 00 (Wu;v; )u2 1 v 2 1 dvdu) f (0) 2;00tf 000 (Wu;v; )u2Z1 2v dv Wu;v; du0sZZs0tf iv (Wu;v; )u41 41vZtf 000 (Wu;v; )u2 v 21dvdu Wu;v; du0;dvdu Hn W(')0tf 0 (Wu;v; )Hn1 (W;(')) K ?;2;adj K ?;2 ' (u; v)dvdu:(19)0We haveZhi@2@2E f (n) (Ws;t; ) p s2 t2 ; y f (n) (y)dy@s@t@s@t RZ@@ p s2 t2 ; y 2 t2 1 s2 f (n) (y)dy@s R @Z@241 41 4 stp s2 t2 ; y f (n) (y)dy2R @Z@ 4 s2 1 t2 1p s2 t2 ; y f (n) (y)dy:@RUsing the integration by parts and the relation@p1 @2p @2 @y 2we obtainhi@2E f (n) (Ws;t; ) @s@t 2s4s21 4t1 2thiE f (n 4) (Ws;t; )hi1E f (n 2) (Ws;t; )1and that proves (19) in the case n 0 (i.e. the case when the test r.v. is F 1). Note also thatihi@ h (n)E f (Ws;t; ) t2 s2 1 E f (n 2) (Ws;t; )@sandWe compute nowi@ h (n)E f (Ws;t; ) t2@this E f (n 2) (Ws;t; ) :1 2hi@2E f (n) (Ws;t; ) h1[0;s] [0;t] ; 'in@s@thii @@2@ h (n) E f (n) (Ws;t; ) h1[0;s] [0;t] ; 'in E f (Ws;t; )h1[0;s] [0;t] ; 'in@s@t@s@ti @hi @2@ h E f (n) (Ws;t; )h1[0;s] [0;t] ; 'in E f (n) (Ws;t; )h1[0;s] [0;t] ; 'in :@t@s@s@t(20)(21)(22)(23)
Ito formula for the two-parameter fractional Brownian motion13On the other hand, using the identity (this is a consequence of the fractional calculus; see [4] for the onedimensional case)Zs0Zt0K ?;2;adj K ?;2 ' (u; v)dvdu h1[0;s]we obtain the relations@h1[0;s]@s[0;t] ; 'i tK ?;2;adj K ?;2 ' (s; v)dv(24)K ?;2;adj K ?;2 ' (u; t)du:(25)0and@h1[0;s]@tZ[0;t] ; 'i[0;t] ; 'i Zs0By combining relations (24), (25) and (21) with (23), we gethi@2E f (n) (Ws;t; ) h1[0;s] [0;t] ; 'in@s@tnhihio s4 1 t4 1 E f (n 4) (Ws;t; ) 2 s2 1 t2 1 E f (n 2) (Ws;t; ) h1[0;s] [0;t] ; 'inZ thi;21 2(n 2)n 1 st E f(Ws;t ) nh1[0;s] [0;t] ; 'iK ?;2;adj K ?;2 ' (u; t)du0Z shi s2 t2 1 E f (n 2) (Ws;t; ) nh1[0;s] [0;t] ; 'in 1K ?;2;adj K ?;2 ' (s; v)dv0Z tZ shi;?;2;adj ?;2(n)n 2 E f (Ws;t ) n(n 1)h1[0;s] [0;t] ; 'iKK ' (s; v)dvK ?;2;adj K ?;2 ' (u; t)du00hi E f (n) (Ws;t; ) nh1[0;s] [0;t] ; 'in 1 K ?;2;adj K ?;2 ' (s; t):ThereforehiE f (n) (Ws;t; ) h1[0;s] [0;t] ; 'inZ sZ t 2 Eu2 1 v 2 1 f (n 2) (Wu;v; )h1[0;u] [0;v] ; 'in dvdu00Z sZ t Eu4 1 v 4 1 f (n 4) (Wu;v; )h1[0;u] [0;v] ; 'in dvdu00Z sZ tZ u21 2(n 2);n 1 Euv f(Wu;v )nh1[0;u] [0;v] ; 'iK ?;2;adj K ?;2 ' (x; v)dx dudv000 its symmetric termZ sZ t Ef (n) (Wu;v; )n(n 1)h1[0;u]Z 0u 0ZK ?;2;adj K ?;2 ' (x; v)dx0 EZ0sZn 2dudv[0;v] ; 'ivK ?;2;adj K ?;2 ' (u; y)dy0tf(n)(Wu;v;)nh1[0;u]n 1 (n)f (Wu;v;[0;v] ; 'i) K ?;2;adj K ?;2 ' (u; v)dudv:(26)0By iterating the duality relation (5) and using (16) and (17), we can proveihE f (n) (Ws;t; ) h1[0;s]n[0;t] ; 'ih n!E f (Ws;t; )Hn (W;i(')) ;(27)
Ito formula for the two-parameter fractional Brownian motionhE f (n 2) (Ws;t;hE f (n 4) (Ws;t;andZsZi) h1[0;s]i) h1[0;s]14hn;'i n!Ef 00 (Ws;t; )Hn (W[0;t]h;niv[0;t] ; 'i n!E f (Ws;t )Hn (W;i(')) ;i;('))(29)tf (n) (Wu;v; )nh1[0;u] [0;v] ; 'in 1 f (n) (Wu;v; ) K ?;2;adj K ?;2 ' (u; v)dudv00Z sZ tf 0 (Wu;v; )Hn 1 (W ; (')) K ?;2;adj K ?;2 ' (u; v)dvdu n!EE(28)0(30)0Taking into account relations (26), (27), (28), (29) and (30), we only need to show thatZ sZ tZ uEu2 1 v 2 f (n 2) (Wu;v; )nh1[0;u] [0;v] ; 'in 1K ?;2;adj K ?;2 ' (x; v)dx dudv0 E0Z0sZ0tf000(Wu;v;)u21 2vdv Wu;v;du Hn (W;('))(31)0(and an analogue for its symmetric term), andZ sZ tEf (n) (Wu;v; )n(n 1)h1[0;u] [0;v] ; 'in 2 dudv0Z u0Z vK ?;2;adj K ?;2 ' (x; v)dxK ?;2;adj K ?;2 ' (u; y)dy00Z ZZ 1[0;s] (u)1[0;t] (v)1[0;u] (u0 )1[0;v] (v 0 )(u;v 0 )2TE Hn2 (W(u0 ;v)2T;('))f 00 (Wu;v; )K ?;2;adj K ?;2 ' (u; v 0 ) K ?;2;adj K ?;2 ' (u0 ; v)To prove the equality (32), we will use the duality relation (10) from De nition (1). We haveZ ZZ 1[0;s] (u)1[0;t] (v)1[0;u] (u0 )1[0;v] (v 0 )(u;v 0 )2T(u0 ;v)2TE Hn 2 (W ; ('))f 00 (Wu;v; ) K ?;2;adj K ?;2 ' (u; v 0 ) K ?;2;adj K ?;2 ' (u0 ; v)Z sZ t dudvE Hn 2 (W ; ('))f 00 (Wu;v; )00Z uZ v?;2;adj ?;2KK ' (x; v)dxK ?;2;adj K ?;2 ' (u; y)dy00and relation (32) follows since similar arguments as above imply thatZ sZ tf (n) (Wu;v; )n(n 1)h1[0;u] [0;v] ; 'in 2 dudvE00Z sZ tdudvE Hn 2 (W ; ('))f 00 (Wu;v; ) : n!0Relation (31) is established similarly.0(32)
Ito formula for the two-parameter fractional Brownian motion415Local timeWe give a brief discussion about the integral representation of the local time of a fractional Brownian sheet.In general, there are two methods to de ne local times for a stochastic process X: the rst one is Berman’sapproach ([3]) based on direct calculations and Fourier analysis, where the local time is de ned as the densityRof the occupation measure t (A) A 1A (Xs )ds; the second method is the Tanaka formula (for processes Xfor which such a formula can be written) where the local time appears as the last term in the decompositionof jXsaj. We have the following situation:for the one-dimensional Brownian motion the two approaches gives the same local time;for the Brownian sheet W the situation changes; we have two di erent local times, the ‘Tanaka formula’local time being the density of the occupation measure (see [6], Chapter 6)Z sZ t(A) 1A (Wu;v )uvdudv;s;t00concerning the fractional Brownian motion B H , the di erence between the two approaches appears evenin the one-parameter case: the Tanaka formula, valid for Skorohod integration, implies the existenceof a local time associated with the weighted occupation measureZ tLt (A) H1A (BsH )s2H 1 ds:0Therefore, since our framework is that of Skorohod integration, it is natural to introduce the local time(Las;t )(s;t)2T;a2R of the fractional Brownian sheet as the density of the occupation measureLs;t;(A) Ls;t (A) Z0sZt1A (Wu;v; )u41 4v1dudv;0de ned for every Borel set A in R. A chaos expansion argument (see e.g. [7]) can be used to show theexistence of the local time. It is also clear that the techniques of the regular case ; 12(see [16]) couldbe adapted to the singular case to obtain a Tanaka-type formula. We will only state the result; the proof isleft to the reader.Proposition 2 For every (s; t) 2 T and a 2 R, it holds thatZ Z211 s tLas;t Ws;t;a Ws;t;aWu;v;a Wu;v;a dWu;v;62 0 0Z sZ tZ sZ t u;v2Wu;v;a u2 1 v 2 1 dvduWu
Ito formula for the two-parameter fractional Brownian motion 1 Itô formula for the two-parameter fractional Brownian motion . to study the problem of stochastic calculus for two-parameter Gaussian processes. The canonical example of such processes is the the fractional . time, including a Tanaka formula. Section 4 describes the extension .
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