This is an announcement for the paper "Representation of Ito integrals by Lebesgue/Bochner integrals" by Qi Lu, Jiongmin Yong and Xu Zhang.
Abstract: In [22], it was proved that as long as the integrand has certain properties, the corresponding It^o 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 can be regarded as a variant of the classical Riesz Representation Theorem, and therefore it will be useful in studying other problems. Some remarkable consequences are presented as well, including a reasonable definition of exact controllability for stochastic differential equations and a condition which implies a Black-Scholes market to be complete.
Archive classification: math.PR math.FA math.OC
Remarks: 26pages
Submitted from: xuzhang@amss.ac.cn
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1007.2969
or