Representation of Itô integrals by Lebesgue/Bochner integrals

• Autores: Qi Lü, Jiongmin Yong, Xu Zhang
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 14, Nº 6, 2012, págs. 1795-1823
• Idioma: inglés
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• Resumen

  • 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.