Representation of Ito integrals by Lebesgue/Bochner integrals
Abbreviated Journal Title
J. Eur. Math. Soc.
Ito integral; Lebesgue integral; Bochner integral; range inclusion; Riesz-type Representation Theorem; STOCHASTIC DIFFERENTIAL-EQUATIONS; BANACH-SPACES; CONTROLLABILITY; OPERATORS; Mathematics, Applied; Mathematics
In , it was proved that as long as the integrand has certain properties, the corresponding Ito 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.
Journal of the European Mathematical Society
"Representation of Ito integrals by Lebesgue/Bochner integrals" (2012). Faculty Bibliography 2010s. 2969.