Representation of Itô integrals by Lebesgue/Bochner integrals

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.