On Asymptotics of Solutions of Stochastic Differential Equations with Jumps

Consider a one-dimensional stochastic differential equation with jumps $$\mathrm{d}X(t)=a(X(t))\mathrm{d}t+\sum _{k=1}^m{b}_k(X(t-))\mathrm{d}{Z}_k(t),$$ where $Z_k,k\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},m\}$ are independent centered Lévy processes with finite second moments. We prove that if coefficient $a(x)$ has certain power asymptotics as $x\to \mathrm{\infty }$ and coefficients $b_k,k\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},m\},$ satisfy certain growth condition then a solution $X(t)$ has the same asymptotics as a solution of $\mathrm{d}x(t)=a(x(t))\mathrm{d}t$ as $t\to \mathrm{\infty }$ a.s.