Stochastic integration of functions with values in a Banach space

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

Studia Mathematica, Tome 166 (2005) no. 2, pp. 131-170

Résumé

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.

Détail

DOI : 10.4064/sm166-2-2

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