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If x ∈ Mcloc (R+, H), then there exists a sequence {τn } of stopping times localizing x for which E supt≥0 |x(t ∧ τn )|2 < ∞. If E|x(t)|2 < ∞ for some t ≥ 0, then E sups

5in 10 N. V. Krylov and B. L. Rozovskii The first important work in this direction was apparently the work of Daletskii [17], where he constructed a Wiener process with an identity covariance operator in a Hilbert space, or, more precisely, in a certain nuclear extension of this space, and defined a stochastic integral. Later, on the basis of the work of Gross [68, 69], Kuo [70] studied a stochastic integral with respect to an abstract Wiener process in a Banach space. The results of Daletskii were also extended in Refs.

In Section 4, the results of Section 3 are applied to the Itˆ o stochastic partial differential equations. 4, where the finitedimensional case is considered, may be of independent interest for some readers. The exposition and notations in these sections are independent of the remainder of the work. 4, while skipping most of the proofs. 5in Stochastic Evolution Equations RozVol 9 2. 1. Introduction The theory of stochastic integration in infinite-dimensional spaces is a broard and rapidly developing area of the theory of stochastic processes.