Stationary and periodic solutions of the operator Riccati equation under a random perturbation (Q1335981)

The paper is a continuation of [author, ibid. 45, No. 2, 255-259 (1993), resp. ibid. 45, No. 2, 239-242 (1993; Zbl 0805.60049)], in which the problem of the existence of bounded and periodic solutions to the operator Riccati equation has been investigated. Here, we establish the existence of stationary and periodic solutions of this equation under random perturbation. It should be noted that the method developed in the above quoted paper is applicable in a more complicated situation: A stationary or periodic perturbation should not necessarily be bounded on \(R\) with probability 1, and, therefore, it is impossible to reduce the problem to the situation where only the local Lipschitz condition is used. In what follows, we study a process periodic with period \(\tau\) (or stationary) all finite-dimensional distributions of which are periodic with period \(\tau\) with respect to time shifts (independent of these shifts).