- Tytuł:
- Régularité Besov des trajectoires du processus intégral de Skorokhod
- Autorzy:
- Lorang, Gérard
- Powiązania:
- https://bibliotekanauki.pl/articles/1287946.pdf
- Data publikacji:
- 1996
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Opis:
- 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 $U_t = ʃ_{0}^{t} uδW$ 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. $t ↦ U_{t} ∈ ℬ_{p,q}^{α}$ 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]).
- Źródło:
-
Studia Mathematica; 1995-1996, 117, 3; 205-223
0039-3223 - Pojawia się w:
- Studia Mathematica
- Dostawca treści:
- Biblioteka Nauki
Artykuł