Régularité Besov des trajectoires du processus intégral de Skorokhod

Lorang, Gérard

Lorang, Gérard

Studia Mathematica, Tome 119 (1996), p. 205-223 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $W_{t} : 0 \leq t \leq 1$ be a linear Brownian motion, starting from 0, defined on the canonical probability space (Ω,ℱ,P). Consider a process $u_{t} : 0 \leq t \leq 1$ belonging to the space $ℒ^{2, 1}$ (see Definition II.2). The Skorokhod integral $U_{t} = ʃ_{0}^{t} u δ W$ is then well defined, for every t ∈ [0,1]. In this paper, we study the Besov regularity of the Skorokhod integral process $t \mapsto U_{t}$ . More precisely, we prove the following THEOREM III.1. (1)If 0 < α < 1/2 and $u \in ℒ^{p, 1}$ with 1/α < p < ∞, then a.s. $t \mapsto U_{t} \in ℬ_{p, q}^{α}$ for all q ∈ [1,∞], and $t \to U_{t} \in ℬ_{p, \infty}^{α, 0}$ . (2) For every even integer p ≥ 4, if there exists δ > 2(p+1) such that $u \in ℒ^{δ, 2} \cap ℒ^{\infty} ([0, 1] \times Ω)$ , then a.s. $t \mapsto U_{t} \in ℬ_{p, \infty}^{1 / 2}$ . (For the definition of the Besov spaces $ℬ_{p, q}^{α}$ and $ℬ_{p, \infty}^{α, 0}$ , see Section I; for the definition of the spaces $ℒ^{p, 1}$ and $ℒ^{p, 2}, p \geq 2$ , see Definition II.2.) An analogous result for the classical Itô integral process has been obtained by B. Roynette in [R]. Let us finally observe that D. Nualart and E. Pardoux [NP] showed that the Skorokhod integral process $t \to U_{t}$ admits an a.s. continuous modification, under smoothness conditions on the integrand similar to those stated in Theorem II.1 (cf. Theorems 5.2 and 5.3 of [NP]).

Publié le : 1996-01-01

Zbl 0858.60042

EUDML-ID : urn:eudml:doc:216252

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     author = {G\'erard Lorang},
     title = {R\'egularit\'e Besov des trajectoires du processus int\'egral de Skorokhod},
     journal = {Studia Mathematica},
     volume = {119},
     year = {1996},
     pages = {205-223},
     zbl = {0858.60042},
     language = {fra},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv117i3p205bwm}
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Lorang, Gérard. Régularité Besov des trajectoires du processus intégral de Skorokhod. Studia Mathematica, Tome 119 (1996) pp. 205-223. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv117i3p205bwm/

Bibliographie

[00000] [BI] M. T. Barlow and P. Imkeller, On some sample path properties of Skorokhod integral processes, in: Séminaire de Probabilités, XXVI, Lecture Notes in Math. 1526, Springer, Berlin, 1992, 1992, 70-80. | Zbl 0761.60047

[00001] [BL] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer, Berlin, 1976. | Zbl 0344.46071

[00002] [B] R. Buckdahn, Quasilinear partial stochastic differential equations without non-anticipation requirement, preprint 176, Humboldt-Universität Berlin, 1988.

[00003] [C] Z. Ciesielski, On the isomorphisms of the spaces $H_{α}$ and m, Bull. Acad. Polon. Sci. 8 (1960), 217-222. | Zbl 0093.12301

[00004] [CKR] Z. Ciesielski, G. Kerkyacharian et B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (1993), 171-204.

[00005] [GT] B. Gaveau et P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel, J. Funct. Anal. 46 (1982), 230-238. | Zbl 0488.60068

[00006] [I1] P. Imkeller, Regularity of Skorohod integral based on integrands in a finite Wiener chaos, Probab. Theory Related Fields 98 (1994), 137-142. | Zbl 0792.60047

[00007] [I2] P. Imkeller, Occupation densities for stochastic integral processes in the second Wiener chaos, ibid. 91 (1992), 1-24.

[00008] [NP] D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands, ibid. 78 (1988), 535-581. | Zbl 0629.60061

[00009] [NZ] D. Nualart and M. Zakai, Generalized stochastic integrals and the Malliavin calculus, ibid. 73 (1986), 255-280. | Zbl 0601.60053

[00010] [PP] E. Pardoux and Ph. Protter, A two-sided stochastic integral and its calculus, ibid. 76 (1987), 15-49.

[00011] [P] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. I, 1976. | Zbl 0356.46038

[00012] [R] B. Roynette, Mouvement Brownien et espaces de Besov, Stochastics Stochastics Rep. 43 (1993), 221-260.

[00013] [S] A. V. Skorokhod, On a generalization of a stochastic integral, Theory Probab. Appl. 20 (1975), 219-233. | Zbl 0333.60060

[00014] [W] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Springer, Berlin, 1984. | Zbl 0546.60054