Stationary solutions of linear systems with additive and multiplicative noise

Abstract

Let , be a (strictly) stationary process with values in which is continuous in pr. and locally integrable. A stationary output y of , is called z-shift if y(t) = θtyo,θ t the shift associated with z. For C=I we generalize results of Bunke and Orey. It is proved that there is a z-shift x on the original pr. space iff there is a stationary x on some pr. space. Necessary and sufficient conditions for existence are given in terms of A and z along with sufficient conditions using the Lyapunov number of z. The proof for the crucial case Reλ(A)=0 is done by a randomization procedure. Conditions for uniqueness (for which Reλ(A)=0 is again the critical case) are given in the form of certain ergodic properties of θ t , The results carry over to the observable case for general C. Finally, we allow A=A(t) to be a stationary process with A(0)∊L 1. Oseledec's multiplicative ergodic theorem enables us to derive a result analogous to that of the case A = const, Reλ(A)≠0