Régularité Besov des trajectoires du processus intégral de Skorokhod

Gérard Lorang

Studia Mathematica (1996)

  • Volume: 117, Issue: 3, page 205-223
  • ISSN: 0039-3223

Abstract

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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 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 ↦ 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 p , q α for all q ∈ [1,∞], and t → U t ∈ ℬ p , ∞ α , 0 . (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 p , ∞ 1 / 2 . (For the definition of the Besov spaces ℬ p , q α and ℬ p , ∞ α , 0 , 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]).

How to cite

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Lorang, Gérard. "Régularité Besov des trajectoires du processus intégral de Skorokhod." Studia Mathematica 117.3 (1996): 205-223. <http://eudml.org/doc/216252>.

@article{Lorang1996,
author = {Lorang, Gérard},
journal = {Studia Mathematica},
keywords = {Besov regularity; Skorokhod integral},
language = {fre},
number = {3},
pages = {205-223},
title = {Régularité Besov des trajectoires du processus intégral de Skorokhod},
url = {http://eudml.org/doc/216252},
volume = {117},
year = {1996},
}

TY - JOUR
AU - Lorang, Gérard
TI - Régularité Besov des trajectoires du processus intégral de Skorokhod
JO - Studia Mathematica
PY - 1996
VL - 117
IS - 3
SP - 205
EP - 223
LA - fre
KW - Besov regularity; Skorokhod integral
UR - http://eudml.org/doc/216252
ER -

References

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  1. [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. Zbl0761.60047
  2. [BL] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer, Berlin, 1976. Zbl0344.46071
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  7. [I1] P. Imkeller, Regularity of Skorohod integral based on integrands in a finite Wiener chaos, Probab. Theory Related Fields 98 (1994), 137-142. Zbl0792.60047
  8. [I2] P. Imkeller, Occupation densities for stochastic integral processes in the second Wiener chaos, ibid. 91 (1992), 1-24. 
  9. [NP] D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands, ibid. 78 (1988), 535-581. Zbl0629.60061
  10. [NZ] D. Nualart and M. Zakai, Generalized stochastic integrals and the Malliavin calculus, ibid. 73 (1986), 255-280. Zbl0601.60053
  11. [PP] E. Pardoux and Ph. Protter, A two-sided stochastic integral and its calculus, ibid. 76 (1987), 15-49. 
  12. [P] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. I, 1976. Zbl0356.46038
  13. [R] B. Roynette, Mouvement Brownien et espaces de Besov, Stochastics Stochastics Rep. 43 (1993), 221-260. 
  14. [S] A. V. Skorokhod, On a generalization of a stochastic integral, Theory Probab. Appl. 20 (1975), 219-233. Zbl0333.60060
  15. [W] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Springer, Berlin, 1984. Zbl0546.60054

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