Régularité Besov des trajectoires du processus intégral de Skorokhod

Let ${\displaystyle \{W_t:0\le t\le 1\}}$ be a linear Brownian motion, starting from 0, defined on the canonical probability space (Ω,ℱ,P). Consider a process ${\displaystyle \{u_t:0\le t\le 1\}}$ belonging to the space ${\displaystyle {ℒ}^{2,1}}$ (see Definition II.2). The Skorokhod integral ${\displaystyle U_t={ʃ}_0^{t}u\delta W}$ is then well defined, for every t ∈ [0,1]. In this paper, we study the Besov regularity of the Skorokhod integral process ${\displaystyle t\mapsto U_t}$. More precisely, we prove the following THEOREM III.1. (1)} If 0 < α < 1/2 and ${\displaystyle u\in {ℒ}^{p,1}}$ with 1/α < p < ∞, then a.s. ${\displaystyle t\mapsto U_t\in {ℬ}_{p,q}^{\alpha }}$ for all q ∈ [1,∞], and ${\displaystyle t\to U_t\in {ℬ}_{p,\infty }^{\alpha ,0}}$. (2)} For every even integer p ≥ 4, if there exists δ > 2(p+1) such that ${\displaystyle u\in {ℒ}^{\delta ,2}\cap {ℒ}^{\infty }([0,1]\times \Omega )}$, then a.s. ${\displaystyle t\mapsto U_t\in {ℬ}_{p,\infty }^{\frac{1}{2}}}$. (For the definition of the Besov spaces ${\displaystyle {ℬ}_{p,q}^{\alpha }}$ and ${\displaystyle {ℬ}_{p,\infty }^{\alpha ,0}}$, see Section I; for the definition of the spaces ${\displaystyle {ℒ}^{p,1}}$ and ${\displaystyle {ℒ}^{p,2},p\ge 2}$, 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 ${\displaystyle t\to U_t}$ 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]).