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Studia Mathematica

1995-1996 | 117 | 3 | 205-223
Tytuł artykułu

Régularité Besov des trajectoires du processus intégral de Skorokhod

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Abstrakty
EN
Let ${W_t : 0 ≤ t ≤ 1}$ be a linear Brownian motion, starting from 0, defined on the canonical probability space (Ω,ℱ,P). Consider a process ${u_t : 0 ≤ t ≤ 1}$ belonging to the space $ℒ^{2,1}$ (see Definition II.2). The Skorokhod integral $U_t = ʃ^{t}_{0} uδW$ is then well defined, for every t ∈ [0,1]. In this paper, we study the Besov regularity of the Skorokhod integral process $t ↦ U_t$. More precisely, we prove the following THEOREM III.1. (1)} If 0 < α < 1/2 and $u ∈ ℒ^{p,1}$ with 1/α < p < ∞, then a.s. $t ↦ U_{t} ∈ ℬ^{α}_{p,q}$ for all q ∈ [1,∞], and $t → U_{t} ∈ ℬ^{α,0}_{p,∞}$. (2)} For every even integer p ≥ 4, if there exists δ > 2(p+1) such that $u ∈ ℒ^{δ,2} ∩ ℒ^∞([0,1]×Ω)$, then a.s. $t ↦ U_t ∈ ℬ^{1/2}_{p,∞}$. (For the definition of the Besov spaces $ℬ^α_{p,q}$ and $ℬ^{α,0}_{p,∞}$, see Section I; for the definition of the spaces $ℒ^{p,1}$ and $ℒ^{p,2}, p ≥ 2$, 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 $t → U_t$ 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]).
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
205-223
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-03-09
poprawiono
1995-08-29
Twórcy
autor
• Institut Elie Cartan, Département de Mathématiques, Université de Nancy I, BP 239, 54506 Vandœuvre-lès-Nancy, Cedex, France
Bibliografia
• [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.
• [BL] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer, Berlin, 1976.
• [B] R. Buckdahn, Quasilinear partial stochastic differential equations without non-anticipation requirement, preprint 176, Humboldt-Universität Berlin, 1988.
• [C] Z. Ciesielski, On the isomorphisms of the spaces $H_α$ and m, Bull. Acad. Polon. Sci. 8 (1960), 217-222.
• [CKR] Z. Ciesielski, G. Kerkyacharian et B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (1993), 171-204.
• [GT] B. Gaveau et P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel, J. Funct. Anal. 46 (1982), 230-238.
• [I1] P. Imkeller, Regularity of Skorohod integral based on integrands in a finite Wiener chaos, Probab. Theory Related Fields 98 (1994), 137-142.
• [I2] P. Imkeller, Occupation densities for stochastic integral processes in the second Wiener chaos, ibid. 91 (1992), 1-24.
• [NP] D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands, ibid. 78 (1988), 535-581.
• [NZ] D. Nualart and M. Zakai, Generalized stochastic integrals and the Malliavin calculus, ibid. 73 (1986), 255-280.
• [PP] E. Pardoux and Ph. Protter, A two-sided stochastic integral and its calculus, ibid. 76 (1987), 15-49.
• [P] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. I, 1976.
• [R] B. Roynette, Mouvement Brownien et espaces de Besov, Stochastics Stochastics Rep. 43 (1993), 221-260.
• [S] A. V. Skorokhod, On a generalization of a stochastic integral, Theory Probab. Appl. 20 (1975), 219-233.
• [W] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Springer, Berlin, 1984.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv117i3p205bwm
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