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

Consider a d-dimensional ItĂ´ 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


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