Representation of Itô integrals by Lebesgue/Bochner integrals

Lü, Qi, Yong, Jiongmin, and Zhang, Xu. "Representation of Itô integrals by Lebesgue/Bochner integrals." Journal of the European Mathematical Society 014.6 (2012): 1795-1823. <http://eudml.org/doc/277306>.

@article{Lü2012,
abstract = {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 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.},
author = {Lü, Qi, Yong, Jiongmin, Zhang, Xu},
journal = {Journal of the European Mathematical Society},
keywords = {Itô integral; Lebesgue integral; Bochner integral; range inclusion; Riesz-type representation theorem; stochastic differential equations; Black-Scholes formula; Itō integral; Lebesgue integral; Bochner integral; range inclusion; Riesz-type representation theorem; stochastic differential equations; Black-Scholes formula},
language = {eng},
number = {6},
pages = {1795-1823},
publisher = {European Mathematical Society Publishing House},
title = {Representation of Itô integrals by Lebesgue/Bochner integrals},
url = {http://eudml.org/doc/277306},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Lü, Qi
AU - Yong, Jiongmin
AU - Zhang, Xu
TI - Representation of Itô integrals by Lebesgue/Bochner integrals
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 6
SP - 1795
EP - 1823
AB - 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 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.
LA - eng
KW - Itô integral; Lebesgue integral; Bochner integral; range inclusion; Riesz-type representation theorem; stochastic differential equations; Black-Scholes formula; Itō integral; Lebesgue integral; Bochner integral; range inclusion; Riesz-type representation theorem; stochastic differential equations; Black-Scholes formula
UR - http://eudml.org/doc/277306
ER -