Title

Representation of Ito integrals by Lebesgue/Bochner integrals

Authors

Authors

Q. Lu; J. M. Yong;X. Zhang

Comments

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Abbreviated Journal Title

J. Eur. Math. Soc.

Keywords

Ito integral; Lebesgue integral; Bochner integral; range inclusion; Riesz-type Representation Theorem; STOCHASTIC DIFFERENTIAL-EQUATIONS; BANACH-SPACES; CONTROLLABILITY; OPERATORS; Mathematics, Applied; Mathematics

Abstract

In [22], it was proved that as long as the integrand has certain properties, the corresponding Ito integral can be written as a (parameterized) Lebesgue integral (or 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 latter 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.

Journal Title

Journal of the European Mathematical Society

Volume

14

Issue/Number

6

Publication Date

1-1-2012

Document Type

Article

Language

English

First Page

1795

Last Page

1823

WOS Identifier

WOS:000311877200003

ISSN

1435-9855

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