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Abstract

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]).