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Régularité Besov des trajectoires du processus intégral de Skorokhod

Tom 117 / 1996

Gérard Lorang Studia Mathematica 117 (1996), 205-223 DOI: 10.4064/sm-117-3-205-223

Streszczenie

Let 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]).

Autorzy

  • Gérard Lorang

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