In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding It\^o integral can be written as a (parameterized) Lebesgue integral (or a 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 later 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.
Cite this article
Qi Lü, Jiongmin Yong, Xu Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1795–1823DOI 10.4171/JEMS/347