1Delft Institute of Applied Mathematics Technical University of Delft P.O. Box 5031 2600 GA Delft, The Netherlands 2Mathematisches Institut I Technische Universität Karlsruhe D-76128 Karlsruhe, Germany
Studia Mathematica, Tome 166 (2005) no. 2, pp. 131-170
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 ${\mit\Phi}:(0,T)\to{\scr L}(H,E)$
with respect to a cylindrical Wiener process $\{W_H(t)\}_{t\in[0,T]}$.
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.
Keywords:
separable real hilbert space real banach space paper construct stochastic integral certain operator valued functions mit phi scr respect cylindrical wiener process construction integral given series expansion terms stochastic integrals certain e valued functions substitute isometry square expectation integral equals radonifying norm operator which canonically associated integrand obtain characterizations class stochastically integrable functions prove various convergence theorems results applied study linear evolution equations additive cylindrical noise general banach spaces example presented linear evolution equation driven one dimensional brownian motion which has weak solution
Affiliations des auteurs :
J. M. A. M. van Neerven
1
;
L. Weis
2
1
Delft Institute of Applied Mathematics Technical University of Delft P.O. Box 5031 2600 GA Delft, The Netherlands
2
Mathematisches Institut I Technische Universität Karlsruhe D-76128 Karlsruhe, Germany
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author = {J. M. A. M. van Neerven and L. Weis},
title = {Stochastic integration of functions with
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J. M. A. M. van Neerven; L. Weis. Stochastic integration of functions with
values in a Banach space. Studia Mathematica, Tome 166 (2005) no. 2, pp. 131-170. doi: 10.4064/sm166-2-2
