Stochastic integration of functions with values in a Banach space

J. M. A. M. van Neerven, and L. Weis. "Stochastic integration of functions with values in a Banach space." Studia Mathematica 166.2 (2005): 131-170. <http://eudml.org/doc/284487>.

@article{J2005,
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.},
author = {J. M. A. M. van Neerven, L. Weis},
journal = {Studia Mathematica},
keywords = {stochastic integration in Banach spaces; Pettis integral; Gaussian covariance operator; Gaussian series; cylindrical noise; convergence theorems; stochastic evolution equations},
language = {eng},
number = {2},
pages = {131-170},
title = {Stochastic integration of functions with values in a Banach space},
url = {http://eudml.org/doc/284487},
volume = {166},
year = {2005},
}

TY - JOUR
AU - J. M. A. M. van Neerven
AU - L. Weis
TI - Stochastic integration of functions with values in a Banach space
JO - Studia Mathematica
PY - 2005
VL - 166
IS - 2
SP - 131
EP - 170
AB - 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.
LA - eng
KW - stochastic integration in Banach spaces; Pettis integral; Gaussian covariance operator; Gaussian series; cylindrical noise; convergence theorems; stochastic evolution equations
UR - http://eudml.org/doc/284487
ER -