Representation of Ito integrals by Lebesgue/Bochner integrals

    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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