m order integrals and generalized Ito's formula the case of a fractional Brownian motion with any

Ivan Nourdin

Description

m-order integrals and generalized Ito's formula; the case of a fractional Brownian motion with any Hurst index Mihai GRADINARU? ,(1), Ivan NOURDIN(1), Francesco RUSSO(2) and Pierre VALLOIS(1) (1) Universite Henri Poincare, Institut de Mathematiques Elie Cartan, B.P. 239, F - 54506 Vandœuvre-les-Nancy Cedex (2) Universite Paris 13, Institut Galilee, Mathematiques, 99, avenue J.B. Clement, F - 93430 Villetaneuse Cedex Abstract: Given an integer m, a probability measure ? on [0, 1], a process X and a real function g, we define the m-order ?-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion BH , for any locally bounded function g, the corresponding integral vanishes for all odd indices m > 12H and any symmetric ?. One consequence is an Ito-Stratonovich type expansion for the fractional Brownian motion with arbitrary Hurst index H ?]0, 1[. On the other hand we show that the classical Ito-Stratonovich formula holds if and only if H > 16 . Key words and phrases: m-order integral, Ito's formula, fractional Brownian motion. 2000 Mathematics Subject Classification: 60H05, 60G15, 60G18.

• path approach


• symmetric integral


• odd indices


• hurst index


• has been


• ?bhs does


• when ?


• integral ∫

Sujets

• Vallois
• Institut Galilée
• Nur al-Din
• Poincaré
• Gr?dinaru
• Cartan
• Has Been

Informations

• Publié par: pefav
• Nombre de lectures: 23
• Langue: English

Extrait

m-orderintegralasdnegenarilezIdo’tˆorsflamuhe;t case of a fractional Brownian motion with any Hurst index Mihai GRADINARU ∗,(1) , Ivan NOURDIN (1) , Francesco RUSSO (2) andPierre VALLOIS (1)

´ (1)Universit´eHenriPoincar´e,InstitutdeMath´ematiquesElieCartan,B.P.239, F - 54506 Vandœuvre-les-Nancy Cedex `

(2)Universite´Paris13,InstitutGalil´ee,Math´ematiques,99,avenueJ.B.Cle´ment, F - 93430 Villetaneuse Cedex

Abstract:Given an integerm, a probability measureνon [0,1], a processXand a real function g, we define them-orderν-integral having as integratorXand as integrandg(X). In the case of the fractional Brownian motionB H , for any locally bounded functiong, the corresponding integral vanishes for all odd indicesm > 21 H and any symmetricνnotartS-oˆtInasihicovencesequecon.On type expansion for the fractional Brownian motion with arbitrary Hurst indexH∈]0,1[. On the other handweshowthattheclassicalItˆo-StratonovichformulaholdsifandonlyifH > 61 .

Key words and phrases:mn.iootnmianworBlanoitcarf,laI,ˆt’ofsroumal-orderintegr

2000 Mathematics Subject Classification:60H05, 60G15, 60G18.

Re´sum´e:Un entierm´eobabilituseredrpu,enemνsur [0,1], un processusXleeltcoirne´tenufeno gtdann´onet´tinfienuo,see´dnν’ordaled´egr-intermayantXeturteraegt´emnicmog(X) comme inte´grand.DanslecasdumouvementbrownienfractionnaireB H , on prouve, pour toute fonction lo- calementborn´eegel’i,qularge´tnpserroceesntdaonepulnn’asinescedirtousloum > 21 H et pour toutes lesmesuressyme´triquesνdelumrofenutneitobone,ncueeqs´onruhcoponivratoo-SteItˆetyp.oCmmce le mouvement brownien fractionnaire d’indice de Hurst quelconque dans ]0,1[. D’autre part, on montre quelaformuled’Itˆo-StratonovichestvalidesietseulementsiH > 1 6 .

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Introduction

The present paper is devoted tom−orderνtni-danItˆo’egralsanofnrno-fsroumal semimartingales.ClassicalItoˆ’sformulaandclassicalcovariationsarefundamentaltoolsof ∗ e-mail: [e-mail]

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stochastic calculus with respect to semimartingales. Calculus involving integratorsXwhich are not semimartingales has been developed essentially in three directions in the last twenty years:

•The case whenXis a Dirichlet process.

•The case whenXis a Gaussian process.

•The case whenXhas paths withp-variation greater than 2.

The implemented techniques for this purpose have been of different natures: the Dirichlet forms approach, the Malliavin (or white noise) calculus approach through the theory of Sko- rohod integral, the Lyons rough path approach and the discretization-regularization approach. It is impossible to list here all the contributors in previous topics; nevertheless we try to sketch some related short history; a survey with a more complete literature could be found in [15].

1. A Dirichlet process may be seen as a natural generalization of a semimartingale: it is constituted by the sum of a local martingale and a zero quadratic variation (instead of a finite variation) process. Such a process is in particular a finite quadratic variation process. Calculus with respect to Dirichlet processes has been developed within two axes. One uses the Dirichlet forms approach, from which the term Dirichlet process was inspired: a fairly complete monography on the subject can be found in [13]. In this framework one can quote for instance [18, 17, 26]. The second approach uses the discretization of the integrals (see e.g. [11, 12, 7]). A counterpart of this approach is the regularization approach (see e. g. [22, 23, 24, 8, 14, 27, 29]). In particular those authors makeuseoftheforwardintegral,whichisanaturalgeneralizationofItˆointegral,and the symmetric integral, which is a natural extension of Stratonovich integral. For those definitions, we refer to section 2.

2. The Skorohod integral, and more generally the Malliavin calculus (see e.g. [20]), has been revealed to be a good tool for considering Gaussian integrators, and in particular fractional Brownian motion. For illustration we quote [6, 1] and [21] for the case ofX being itself a Skorohod integral.

3. The rough path approach has been performed by T. Lyons [16], and continued by several authors; among them, [5] has adapted this technique to the the study of SDEs driven by fractional Brownian motion. The regularization approach has been recently continued by [9, 15] to analyze calculus with respect to integrands whosen-variation is greater than 2, developing the notion of n-covariation. In particular, [9] introduces the notion of 3-variation (or cubic variation) of a process, denoted by [X, X, X].

We come back now to the main application of this paper, that is fractional Brownian motion. This process, which in general is not a semimartingale, has been studied intensively in stochas- tic analysis and it is considered in many applications, e.g. in hydrology, telecommunications, fluidodynamics, economics and finance.

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Recall that a mean zero Gaussian processX=B H is a fractional Brownian motion with Hurst indexH∈]0,1[ if its covariance function is given by K H (s, t= 1 )( 2 |s| 2H +|t| 2H − |s−t| 2H ),(s, t)∈R 2 .(1.1) An easy consequence of that property is that E(B tH −B sH ) 2 =|t−s| 2H .(1.2) WhenH= 2 1 ,B H is the classical Brownian motion. It is well-known thatB H is a semi- martingale if and only ifH= 1 2  the other hand, if. OnH > 1 2 ,B H is a zero quadratic 1 variation process, therefore (trivially) also a Dirichlet process. As we said, ifH≥ 2 ,B H is afinitequadraticvariationprocess,thereforeanItˆo’sformulainvolvingsymmetricintegrals holds, and it can be deduced from [23, 11]. Iffis of class C 2 , we have f(B tH ) =f(B 0 H ) + Z 0 t f 0 (B sH )d ◦ B sH .(1.3) IfH > 2 1 , [B H , B H ] vanishes and the symmetric integral R 0 t f 0 (B sH )d ◦ B sH coincides with the forward integral R 0 t f 0 (B sH )d − B sH . Settingf(x) =x 2 (1.3) says that , (B tH ) 2 = (B 0 H Z 0 t B sH d ◦ B sH .(1.4) ) 2 + 2 IfH < 2 1 the forward integral R 0 t B sH d − B sH  In fact,does not exist, but (1.4) is still valid. using the identity 2 (B sH +ε  () =B sH ) 2 + 2 B sH +ε 2 +B sH (B sH +ε −B sH ),(1.5) integrating from zero totboth members of the equality, dividing byεand using the definition of symmetric integral, we can immediately see that (1.4) holds for any 0< H <1. The natural question which arises is the following: is (1.3) valid for any 0< H <1? The In answer is no. reality, takingf(x) =x 3 , similarly to (1.5), we can expand as follows H s ) 2 H (B s+ε −B sH ) 3 (B sH +ε ) 3 = (B sH ) 3 + 3  (B sH +ε ) 2 2 +(B H (B s+ε −B sH )− 2 . Proceeding as before, (B tH ) 3 could be expanded as H (B tH ) 3 = (B 0 H ) 3 + 3 Z 0 t (B sH ) 2 d ◦ B sH − [B , B H , B H ] t ; (1.6) 2 moreover previous symmetric integral will exist if and only if [B H , B H , B H ] exists. In reality, that object exists if and only ifH > 6 1 : in that case the mentioned cubic variation even vanishes. This point comes out as a consequence of Theorem 4.1 2. when

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