In this paper we construct a theory of stochastic integration of processes with values in $\mathcal{L}(H,E)$, where $H$ is a separable Hilbert space and $E$ is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an $H$-cylindrical Brownian motion. Our approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of $\mathcal{L}(H,E)$-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the It\^{o} isometry, the Burkholder--Davis--Gundy inequalities, and the representation theorem for Brownian martingales.Comment: Published at http://dx.doi.org/10.1214/009117906000001006 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equationwhere A generates an analytic C 0 -semigroup on a UMD Banach space E and W H is a cylindrical Brownian motion with values in a Hilbert space H . We prove that if the mappings F : [0, T ] × E → E and B : [0, T ] × E → L(H, E) satisfy suitable Lipschitz conditions and u 0 is F 0 -measurable and bounded, then this problem has a unique mild solution, which has trajectories in C λ ([0, T ]; D((−A) θ ))) provided λ 0 and θ 0 satisfy λ + θ < 1 2 . Various extensions are given and the results are applied to parabolic stochastic partial differential equations. addresses: J.M.A.M.vanNeerven@tudelft.nl (J.M.A.M. van Neerven), mark@profsonline.nl (M.C. Veraar), Lutz.Weis@math.uni-karlsruhe.de (L. Weis).

Abstract. Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ : (0, T ) → L(H, E) with respect to a cylindrical Wiener process {W H (t)} t∈ [0,T ] . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is canonically associated with the integrand. We obtain characterizations for the class of stochastically integrable functions and prove various convergence theorems. The results are applied to the study of linear evolution equations with additive cylindrical noise in general Banach spaces. An example is presented of a linear evolution equation driven by a one-dimensional Brownian motion which has no weak solution.In this paper we construct a theory of stochastic integration of operatorvalued functions with respect to a cylindrical Wiener process. The range space of the operators is allowed to be an arbitrary real Banach space E. A stochastic integral of this type can be used for solving the linear stochastic Cauchy problemHere, A is the infinitesimal generator of a strongly continuous semigroup {S(t)} t≥0 of bounded linear operators on E, the operator B is a bounded linear operator from a separable real Hilbert space H into E, and {W H (t)} t∈ [0,T ] 2000 Mathematics Subject Classification: Primary 60H05; Secondary 28C20, 35R15, 47D06, 60H15.Key words and phrases: stochastic integration in Banach spaces, Pettis integral, Gaussian covariance operator, Gaussian series, cylindrical noise, convergence theorems, stochastic evolution equations.The first named author gratefully acknowledges the support by a 'VIDI subsidie' in the 'Vernieuwingsimpuls' programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1-1).[131] It is well known that the integral on the right hand side can be interpreted as an Itô stochastic integral if E is a Hilbert space. A comprehensive theory of abstract stochastic differential equations in Hilbert spaces is presented in the monograph by Da Prato and Zabczyk [5]. More generally the integral can be defined for spaces E with martingale type 2. This has been worked out by Brzeźniak [2]. Examples of martingale type 2 spaces are Hilbert spaces and the Lebesgue spaces L p (µ) with p ∈ [2, ∞).In both settings, the integral is defined for step functions first, and then for general functions by a limiting argument. Such an argument depends on the availability of a priori estimates for the integrals of the approximating step functions, and the martingale type 2 property is precisely designed to provide such estimates.Without special assumptions on the geometry of the underl...

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation. (Z. Brzeźniak), j.m.a.m.vanneerven@tudelft.nl (J.M.A.M. van Neerven), m.c.veraar@tudelft.nl (M.C. Veraar), lutz.weis@mathematik.uni-karlsruhe.de (L. Weis).