# Representation of Itô integrals by Lebesgue/Bochner integrals

Qi Lü; Jiongmin Yong; Xu Zhang

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 6, page 1795-1823
- ISSN: 1435-9855

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topLü, 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 -

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