Title

Representation Of Itô Integrals By Lebesgue/Bochner Integrals

Keywords

Bochner integral; Itô integral; Lebesgue integral; Range inclusion; Riesz-type Representation Theorem

Abstract

In [22], 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 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. © 2012 European Mathematical Society.

Publication Date

1-1-2012

Publication Title

Journal of the European Mathematical Society

Volume

14

Issue

6

Number of Pages

1795-1823

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.4171/JEMS/347

Socpus ID

84868261563 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84868261563

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