Geodesic
Representation of Itô integrals by Lebesgue/Bochner integrals
Journal of the European Mathematical Society, Tome 14 (2012) no. 6, pp. 1795-1823 Cet article a éte moissonné depuis la source EMS Press

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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.
DOI : 10.4171/jems/347
Classification : 60-XX, 00-XX
Keywords: Itô integral, Lebesgue integral, Bochner integral, range inclusion, Riesz-type representation theorem 
@article{JEMS_2012_14_6_a2,
     author = {Qi L\"u and Jiongmin Yong and Xu Zhang},
     title = {Representation of {It\^o} integrals by {Lebesgue/Bochner} integrals},
     journal = {Journal of the European Mathematical Society},
     pages = {1795--1823},
     year = {2012},
     volume = {14},
     number = {6},
     doi = {10.4171/jems/347},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/347/}
}
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Qi Lü; Jiongmin Yong; Xu Zhang. Representation of Itô integrals by Lebesgue/Bochner integrals. Journal of the European Mathematical Society, Tome 14 (2012) no. 6, pp. 1795-1823. doi: 10.4171/jems/347

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