Solutions of affine stochastic functional differential equations in the state space

Markus Riedle

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 05-3

MSC 2000

60H20 Stochastic integral equations

60G15 Gaussian processes

60J35 Transition functions, generators and resolvents

34K50 Stochastic delay equations

Abstract
In this article we consider solutions of affine stochastic functional differential equations. The drift of these equations is specified by a functional defined on a general function space B which is only described axiomatically. The solutions are reformulated as stochastic processes in the space B. By representing such a process in the bidual space of B we establish that the transition functions of this process form a generalized Gaussian Mehler semigroup on B. Thus the process is characterized completely on B since it is Markovian. Moreover we derive a sufficient and necessary condition on the underlying space B such that the transition functions are even an Ornstein-Uhlenbeck semigroup. We exploit this result to associate a Cauchy problem in the function space B to the finite-dimensional functional equation.