## On the asymptotic degeneration of systems of linear inhomogeneous stochastic differential equations.(Ukrainian, English)Zbl 1199.60213

Teor. Jmovirn. Mat. Stat. 76, 38-44 (2007); translation in Theory Probab. Math. Stat. 76, 41-48 (2008).
The author deals with the following system of differential equations $dx(t)=(A_0x(t)+f_0(t))dt+\sum_{k=1}^{m}(A_kx(t)+f_k(t))dW_k(t),$ where $$A_k$$ are $$n\times{n}$$ matrices; $$f_k(t)=(f_{k1}(t),\dots,f_{kn}(t)), t\geq0$$, are vector functions; $$W_k(t),t\geq0$$, are independent one-dimensional Wiener processes; $$x(t)=(x_1(t),\dots,x_n(t)),t\geq0$$, is a solution to the equations.
Let $$H_s^t$$ be a stochastic semigroup of nondegenerate operators such that $$dH_s^t=A_0H_0^tdt+\sum_{k=1}^mA_kH_s^tdW_k(t), H_s^s=I,s\leq t.$$ It is assumed that the semigroup $$H_s^t$$ is stable with probability one. A solution $$x(t),t\geq0,$$ to the system of equations can be represented in the form $x(t)=H_0^tx+\int_0^t(H_u^t)\left(f_0(u)-\sum_{k=1}^mA_kf_k(u)\right)du+\sum_{k=1}^mH_0^t\int_0^t(H_0^u)^{-1}f_k(u)dW_k(u).$ The author obtains sufficient conditions for the convergence to zero, in probability as well as pathwise, of solutions $$x(t)$$ of the indicated system of linear inhomogeneous stochastic differential equations.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness
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