## Bounded and periodic solutions of affine stochastic differential equations.(English)Zbl 0623.60075

Consider a d-dimensional Itô stochastic differential equation (SDE) of the type $(1)\quad dx(t) = (A(t)x(t)+f(t))dt+\sum^{m}_{j=1}(B_ j(t)x(t)+h_ j(t))dw_ j(t),\quad t\in {\mathbb{R}},$ with bounded coefficients, which the author calls an affine system of SDE’s. Solutions are defined in the a.e. sense. Consider also the corresponding (so- called) linear system of SDE’s $(2)\quad dy(t) = A(t)x(t)dt+\sum^{m}_{j=1}B_ j(t)y(t)dw_ j(t),\quad t\in {\mathbb{R}}.$ The paper proves that: (a) If the zero solution of (2) is exponentially stable in mean square, then the system (1) has a unique bounded solution (bounded in the mean square sense). (b) If, further, the coefficients are periodic with period $$\theta$$, this solution is periodic with period $$\theta$$ (its finite-dimensional distributions are invariant to $$\theta$$ time shifts) and any weakly $$\theta$$-periodic (i.e., $$\theta$$-periodic in the mean square sense) solution is a.e. the solution referred to in (a).
Reviewer: C.A.Braumann

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93E15 Stochastic stability in control theory