Stationary in wide sense solutions of difference equations in the Banach space. (Ukrainian, English) Zbl 1150.60014

Gorodnij, M. F.

Teor. Jmovirn. Mat. Stat. 74, 27-33 (2006); translation in Theory Probab. Math. Stat. 74, 29-35 (2007).

Let \((X,\|\cdot\|_{X})\) be a complex separable Banach space, let \(\eta=\{\eta_{n}, n\in Z\}\) be a stationary sequence of \(X\)-valued random elements, let \(A:\;D\subset X\to X\) be a closed operator, and let \(\{A_{n}, n\in Z\}\) be a sequence of operators from \(L(X)\) such that \(\sum_{n\in Z}\| A_{n}\|_{L(X)}<+\infty\). The author proves existence and uniqueness of stationary solution of difference equation \(A\xi_{n}=\sum_{k\in Z}A_{k}\xi_{n+k}+\eta_{n}\), \(n\in Z\), for any stationary sequence \(\eta\). The stability of stationary solution of the considered difference equation with respect to small disturbances of operator coefficients is established.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60G10	Stationary stochastic processes
39A10	Additive difference equations
47A50	Equations and inequalities involving linear operators, with vector unknowns

Keywords:

stationary in wide sense solutions; difference equations in the Banach space; operator coefficients