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Weak characterizations of stochastic integrability and Dudley’s theorem in infinite dimensions. (English) Zbl 1316.60087

This article is devoted to weak characterizations of stochastic integrability with respect to a cylindrical Brownian motion \(W\) in an infinite-dimensional Banach space \(E\).
First, the following general representation theorem is established: Any strongly \(F_T\)-measurable \(\xi:\Omega\to E\) can be written as \(\xi= \int^T_0 \phi\,dW\), for some stochastically integrable process \(\phi\).
Then, the main result concerns a reflexive space \(E\), a Hilbert-space \(H\), an \(H\)-strongly measurable adapted process \(\phi: (0,T)\times\Omega\to L(H,E)\) which is weakly in \(L^0(\Omega, L^2((0,T),H))\), and a process \(\xi:(0,T)\times\Omega\to E\) having bounded paths.
The authors prove that if for any \(x^*\in E^*\) and \(t\leq T\) \[ \langle\xi(t), x^*\rangle= \int^t_0 \phi^* x^*dW_H \;\text{ a.s., \;\;then }\xi(t)= \int^t_0 \phi\,dW_H \;\text{ a.s.} \] and \(\xi(t)\) is a local martingale.

MSC:

60H05 Stochastic integrals
60B11 Probability theory on linear topological spaces
60F99 Limit theorems in probability theory
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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