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Régularité Besov des trajectoires du processus intégral de Skorokhod

Gérard Lorang (1996)

Studia Mathematica

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 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 p , 1 with 1/α < p < ∞, then a.s. t U t p , q α for all q ∈ [1,∞], and t U t p , α , 0 . (2) For every even integer p ≥...

Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

Byoung Soo Kim, Dong Hyun Cho (2017)

Czechoslovak Mathematical Journal

Let C [ 0 , t ] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [ 0 , t ] , and define a random vector Z n : C [ 0 , t ] n + 1 by Z n ( x ) = x ( 0 ) + a ( 0 ) , 0 t 1 h ( s ) d x ( s ) + x ( 0 ) + a ( t 1 ) , , 0 t n h ( s ) d x ( s ) + x ( 0 ) + a ( t n ) , where a C [ 0 , t ] , h L 2 [ 0 , t ] , and 0 < t 1 < < t n t is a partition of [ 0 , t ] . Using simple formulas for generalized conditional Wiener integrals, given Z n we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra 𝒮 . Finally, we express the generalized analytic conditional Feynman...

Representation of Itô integrals by Lebesgue/Bochner integrals

Qi Lü, Jiongmin Yong, Xu Zhang (2012)

Journal of the European Mathematical Society

In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding Itô integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later...

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