Stochastic Integration with respect to Gaussian Processes

Transcription

1 C. R. Acad. ci. Paris, t. 2, érie I, p., 2 - PXMA.TEX - Probabilités/Probability (/ tochastic Integration with respect to Gaussian Processes Laurent Decreusefond a, a E.N..T., rue Barrault, 5 Paris cedex Tél : 5 8 5, Fax : Laurent.Decreusefond@enst.fr (Reçu le jour mois année, accepté le jour mois année Abstract. Résumé. We construct a tratonovitch-orohod-lie stochastic integral for general Gaussian processes. We study its sample-paths regularity one of its numerical approximating schemes. We also analyze the way it is transformed by an absolutely continuous change of probability we give an Itô formula. c 2 Académie des sciences/éditions scientifiques et médicales Elsevier A Nous construisons une intégrale stochastique du type tratonovitch-orohod, pour les processus gaussiens généraux. Nous montrons qu elle peut être approchée par des sommes de type tratonovitch et nous établissons sa régularité trajectorielle. Nous étudions aussi la façon dont elle se transforme lors d un changement absolument continu de probabilité. Nous montrons enfin que la formule d Itô-tratonovitch est vérifiée. c 2 Académie des sciences/éditions scientifiques et médicales Elsevier A Version française abrégée De nombreux auteurs se sont attachés à construire les fondements d un calcul stochastique pour le mouvement brownien fractionnaire. Les diverses approches peuvent être séparées en deux principales catégories : celles qui reposent sur les propriétés trajectorielles de ce processus [, 8, ] (notamment la continuité au sens de Hölder de ses trajectoires et celles qui utilisent sur son caractère gaussien. Ces dernières sont basées sur la représentation où est un noyau déterministe. L expérience des calculs faits pour le mouvement brownien fractionnaire montre qu il est ardu de travailler directement avec des hypothèses sur le noyau mais qu il est plus aisé et plus informartif de travailler avec les propriétés de l opérateur intégral canoniquement associé à ce noyau. Pour un noyau déterministe qui satisfait l hypothèse I (voir plus bas, on considère le processus!. Compte-tenu des résultats de [], on sait que a une version à trajectoires p.s. continues. Par conséquent, l espace de Wiener sur lequel nous travaillons, est #$$% (**-,./ l espace de Cameron- Martin est 2 (**5 et est la probabilité sur # qui fait du processus canonique, un processus Note présentée par -2(-/FLA c 2 Académie des sciences/éditions scientifiques et médicales Elsevier A. Tous droits réservés.

2 gaussien de même loi que Compte-tenu des résultats antérieurs obtenus sur le mouvement brownien fractionnaire, une définition raisonnable de l intégrale stochastique de par rapport à est est ici l adjoint dans ( 5 Ceci exige que appartienne au domaine de et que soit p.s. un opérateur à trace. n définit ensuite une intégrale de type tratonovitch par limite de sommes discrètes qui coïncide avec la précédente lorsque est nucléaire, mais qui possède un domaine plus facile à caractériser. n peut aussi, grâce aux inégalités de Meyer, établir les hypothèses sur l intégr sous lesquelles on a une inégalité maximale pour la partie divergence, voir les théorèmes.2 et.. Les formules de composition des intégrales stochastiques par un shift absolument continu de l espace de Wiener, montrent que la première de ces intégrales se transforme formellement comme une semi-martingale lors d un changement de probabilité absolument continue, voir le théorème.. Pour terminer, nous établissons la formule d Itô-tratonovitch.. Introduction In the past few years, several papers were devoted to the stochastic analysis of fractional Brownian motion. A natural problem is now to extend the theory of stochastic calculus, developed for fbm, to a larger class of processes. ample-paths constructions, as those devised in [8,, ], can be applied to processes with rough paths but this is not the way we chose. As long as fbm is concerned, the other approach to construct a stochastic integral, is based on its representation as a stochastic integral with respect to a stard Brownian :! It is clear that the regularity of the ernel plays a ey role in a tentative definition of a stochastic calculus with respect to The experience of the past wors for fractional Brownian motion shows that it is simpler more insightful to wor with the property of the linear map associated to the ernel than with the properties of the ernel itself (see [2] for such an approach. 2. Definition Let be a Hilbert-chmidt map from * into itself of the form 2 For any $(!#%$ we introduce the lobodetzi space (see [, ]: 5 8 : (* -, /.2. Ä@ - D d d EF$HG A@ <> CB (* KJL For any I$( N P P ( Furthermore, M is continuously embedded in RQT (**5 when # in UWVX-YZ@\[] when ^/ We introduce the following assumption. Moreover, B(a` is triangular, i.e., for Let ( It is shown in [] that under hypothesis I, there exists a measurable version of which has a.s. b@dc -Hölder continuous sample-paths, for any c# Hence, the Wiener space is # % (,. the Cameron-Martin space is * the probability on # under which HYPTHEI I. There exists _ such that is continuous, one-to-one, from ( into the canonical process has the law of As usual, for e a separable Hilbert space, f Ye denote the space of e valued r.v. which are once Gross-obolev differentiable the derivative of which belongs to -#Zg **-,]e When I P[h( the map defined by i j2 is a continuous map from ( to R For C` \[lh we will use the following assumption: 2

3 %2 ` 2IH %2, Vb HYPTHEI II. For d \[lh we assume that is continuous from * into ( is C` a densely defined, closable operator from ( into itself. We denote by its closure by its adjoint. We furthermore assume that Vb is dense in (**5 Remar. Hypothesis I II are satisfied for the ernel corresponding to fbm (with stationary increments or not, cf. []. For the ernel these hypothesis are also satisfied for some ( if there exists /% such that is a -Hölder continuous function I% This is the generalization of the fbm introduced in []. Note that in all the subsequent theorems, the hypothesis are all satisfied by For the sae of simplicity, we will spea of the domains of independently of the position of with respect to P[h It must be clear that for I P[h( V Vb ( We denote by the adjoint of on ( 5 by the valued Gross-obolev derivative of the stochastic calculus of variations (usually denoted V V V% see [, 5]. The adjoint of with respect to is called divergence is denoted by its domain is We denote by V% the set of processes belonging a.s. to such that belongs to V% represents the set of processes in V% such that V is P-a.s. a trace class operator. Following the definition of a stochastic integral with respect to fbm first introduced in [], then in [] also in [2] for some Gaussian processes, we set: DEFINITIN 2.. Assume that Assumption I II hold. For we define the stochastic integral on ( - of with respect to by d! a]l2y It is useful to note that a]l Y provided that one of these two terms exists. Another approach is to follow the tratonovitch integration principles. We first to approximate by a linear / interpolation to obtain an approximation of %$ # then obtain an approximation of This leads to consider: %$ B %$( 2 * We say that is tratonovitch integrable when the family B converges in probability as The d The proofs of the following results if not given here, are established in a paper in preparation [5]. THEREM 2.2. Assume that Hypothesis I II hold for some H \[lh Assume furthermore that there exists/ such that is continuous from * B. into B. If a` N belongs to f E 58 W@:2 CB. there exists a measurable integrable process, denoted by such that, for almost any2 > 5 25 %2 ^ 5 < ( (@Q Moreover, C8 5 EA d2degf (2 J d2 (@Q Furthermore, is tratonovith integrable the limit of B is. THEREM 2.. Assume that Hypothesis I holds for % P[h Assume furthermore that is continuous from ( into for some Z YZ@$\[lh a` If P belongs to f E KL8 W@M2 * then there exists a measurable integrable process, denoted by such that, for almost N8 any2 K2 %2 C` a` 5 ( limit is denoted by -.- CB P < - G-d

4 f Z %2 8 8 Vb 5 5 %2 H C8 Furthermore, is tratonovith integrable the limit of B is B Moreover, DEGF J8 8 EA d2 d2 ( C` Remar 2. As in [], it follows that if satisfies the hypothesis of the previous theorems if d -.- d coincide.. Regularity of the trajectories is a trace class operator then, the two integrals - We now turn to the regularity of the stochastic integral. As will be seen, is always continuous from Cj! N into **5 We denote by C` its norm. / THEREM.. For any ( \[lh assume that assumptions I II hold. Let belong to B The process a` ( admits a modification with -Hölder continuous paths for any K Z we have the maximal inequality: B >DEFÖ N N Q Proof. Note that since is triangular, we have for any 2 ince the divergence operator is continuous from f ( in -# we have 8 E Y EA E (H! 2! (5 < Under assumption II, is continuous from ( into thus a` is continuous from j# N! a` into ( We have: 2H a` C8 5P 5 * E E EA Q Q According to Theorem 2. of [2] with 8 / we have 5 N 8 < Q%$ Q imple computations give that P@ the maximal inequality follows by a functional analytic version of the Kolmogorov criterion Q%$ []. P THEREM.2. For any \[h(* assume that assumption I holds. Let P belong to f with The process ( * admits a modification P[] -Hölder continuous paths for any % N8 we have the maximal inequality: B >DEGFR T a` 8 J82 Proof. The proof of Theorem.2 is identical until Eqn. (5. ince L\[lh it is clear that is continuous from **5 into thus that a` is continuous from j in **5 ince C` C` is continuously embedded in QT it follows that C` 2 a` Cj is continuously embedded in N ince a` P belongs to f 2 Hölder inequality implies that 8 :@ C` with/ A similar inequality is valid for the other term of (5 the proof is then complete.

5 $. Change of variable Another feature of this stochastic integral is that its composition with absolutely continuous shifts of the Wiener space is simple to compute. THEREM.. Let L be such that belongs to f for some. Let be such that - belong to V% are a.s. trace class operators. Then, - d d A Proof..- According to Proposition B..2 B..8 of [], we have a]l2 - - %a]ly - *2 2 a]l2.- The proof is completed by substituting the latter equation into the former. For deterministic adapted, this means that the law of the process \@ 2 - ( I under is identical to the -law of the process EI$ - 5. Itô s formula We are now interested in non-linear transformations of Itô-lie processes: A - d a sufficiently regular The ey remar is that the triangularity of the ernel implies that on ( THEREM 5.. Let be a % -function suppose that V is a smooth-cylindric process, i.e., D2 Y ( where is a rapidly decreasing function from. for any belongs to Vb Let - d Then, j- belongs to into. belongs to V for 2 j - d ( Proof. [etch of the proof] The proof follows the lines of the method initiated in []. Let be an. -valued smooth-cylindric functional, E Ä@ E H j # * W@. $ %$ In view of the definition of the stochastic integral with respect to of the triangularity F - d where %, thus! dh E j! ( EA j - EA j 2 dh # EA 2 j EA j! 2 d*h dh 5

6 $ E % j- It is then clear that c $ converges to E j! T E j- ince j j is similar computations show that c $ converges to as c goes to ne can then use the fundamental lemma of calculus we Hobtain: E Ä@ ( EA j! d EA j! d H H j-.- H EA j! d EA 2 j d Equation ( follows by identification. E By density of the smooth-cylindric functionals, one has the more general result: THEREM 5.2. Assume that we have, either * into B. C` N belongs to f or Hypothesis I II hold for some P[h there exists/ such that is continuous from CB. Hypothesis I holds for I \[lh is continuous CB. from * into for some d a` /@ P[h P belongs to f * Then, for j- belongs to V% 8 ( is true. The proof boils down to prove that 8 J8 J where * P/ h when \[lh CB. * when LP[h This is done by repetitive use of the previous theorems 2.2, 2.,..2. References [] E. Alòs, J.A. livier León, D. Nualart, tochastic tratonovitch calculus for fractional Brownian motion with Hurst parameter less than, Taiwanese Journal of Mathematics 5 (2, no., 2. [2] E. Alòs,. Mazet, D. Nualart, tochastic calculus with respect to Gaussian processes, Annals of Probability (22, To appear. [] A. Benassi, P. Bertr,. Cohen, J. Istas, Identification d un processus gaussien multifractionnaire avec des ruptures sur la fonction d échelle, C. R. Acad. ci. Paris ér. I Math. 2 (, no. 5, 5. [] L. Decreusefond, Regularity properties of some stochastic Volterra integrals with singular ernel, Potential Analysis (22,. [5], tochastic calculus for Volterra processes, In preparation, 22. [] L. Decreusefond A.. Üstünel, tochastic analysis of the fractional Brownian motion, Potential Anal. (, no. 2, 2. [] D. Feyel A. de La Pradelle, n fractional Brownian processes, Potential Anal. (, no., [8] M. Gradinaru, F. Russo, P. Vallois, Generalized covariations, local time tratonovitch Itô s formula for fractional Brownian motion with Hurst index, Tech. Report, Université Paris XIII, 2. [] T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana (8, no. 2, 25. [] D. Nualart, The Malliavin calculus related topics, pringer Verlag, 5. [] D. Nualart M. Zaai, n the relation between the tratonovich gawa integrals, Ann. Probab. (8, no., 5 5. [2] J. Tambača, Estimates of the obolev norm of a product of two functions, J. Math. Anal. Appl. 255 (2, no.,. [] A.. Üstünel, The Itô formula for anticipative processes with nonmonotonous time scale via the Malliavin calculus, Probability Theory related fields (88, 2 2. [] A. üleyman Üstünel M. Zaai, Transformation of measure on Wiener space, pringer-verlag, Berlin, 2. [5] A.. Üstünel, An introduction to analysis on Wiener space, Lectures Notes in Mathematics, vol., pringer- Verlag, 5. [] M. Zähle, Integration with respect to fractal functions stochastic calculus, I, Probability Theory Related Fields (8, no.,.