Solutions of stochastic delay equations in spaces of continuous functions

Markus Riedle , Jan M. A. M. vanNeerven

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 18-2005, 11

MSC 2000

34K50 Stochastic delay equations

34F05 Equations and systems with randomness

47D06 One-parameter semigroups and linear evolution equations

60H05 Stochastic integrals

Abstract
We present a semigroup approach to stochastic delay equations with linear drift and additive noise in the space of continuous functions $C[-h,0]$. We represent the solution as a $C[-h,0]$-valued process arising from a stochastic weak$\s$-integral in the bidual $C[-h,0]^{**}$ and show how this process can be interpreted as a mild solution of an associated stochastic abstract Cauchy problem. We obtain a necessary and sufficient condition guaranteeing the existence of an invariant measure.