LYAPUNOV EXPONENTS OF LINEAR STOCHASTIC FUNCTIONAL-DIFFERENTIAL EQUATIONS DRIVEN BY SEMIMARTINGALES .1. THE MULTIPLICATIVE ERGODIC-THEORY

We consider a class of stochastic linear functional differential syste ms driven by semimartingales with stationary ergodic increments, We al low smooth convolution-type dependence of the noise terms on the histo ry of the state. Using a stochastic variational technique we construct a compactifying stochastic semiflow on the state space. As a necessar y ingredient of this construction we prove a general perfection theore m for cocycles with values in a topological group (Theorem 3.1). This theorem is an extension of a previous result of de Sam Lazaro and Meye r (cf. [7], Theorem 1, p. 40). A multiplicative Ruelle-Oseledec ergodi c theorem then gives the existence of a discrete Lyapunov spectrum and a saddle-point property in the hyperbolic case.