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In [22], it was proved that as long as the integrand has certain properties,the corresponding It\^o integral can be written as a (parameterized) Lebesgueintegral (or a Bochner integral). In this paper, we show that such a questioncan be answered in a more positive and refined way. To do this, we need tocharacterize the dual of the Banach space of some vector-valued stochasticprocesses having different integrability with respect to the time variable andthe probability measure. The later can be regarded as a variant of theclassical Riesz Representation Theorem, and therefore it will be useful instudying other problems. Some remarkable consequences are presented as well,including a reasonable definition of exact controllability for stochasticdifferential equations and a condition which implies a Black-Scholes market tobe complete.

Representation of It\^o Integrals by Lebesgue/Bochner Integrals