Stationary and periodic solutions of a stochastic difference equation in a Banach space. (English. Russian original) Zbl 0746.60070

Dorogovtsev, A. Ya.

Theory Probab. Math. Stat. 42, 39-46 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 35-42 (1990).

Consider the difference equation \[ Ax(n) = x(n+1) - 2x(n) + x(n-1) + \epsilon (n),\quad n\in \mathbb{Z},\tag{\(*\)} \] in a Banach space \(B\), where the operator \(A:B\to B\) is bounded, and \(\{ \epsilon (n),\;n\in \mathbb{Z}\}\) is a stationary \(B\)-valued stochastic process with \(E\|\epsilon (0)\| < \infty\). Equation \((*)\) has a stationary (in restricted sense) solution iff the interval \([-4,0]\) belongs to the resolvent set of the operator \(A\). Some similar results are obtained in the case when the operator \(A\) is unbounded and in the case when the right-hand side of \((*)\) is enlarged by some nonlinear term. The existence of periodic solutions is also investigated.

Reviewer: D.Bobrowski (Poznań)

MSC:

60H99	Stochastic analysis
39A10	Additive difference equations

Keywords:

resolvent set of an operator in a Banach space; stationary solution; difference equation; existence of periodic solutions