On the Asymptotics of Solutions of Stochastic Differential Equations with Jumps

Consider a one-dimensional stochastic differential equation with jumps

\(dX\left(t\right)=a\left(X\left(t\right)\right)dt+\sum_{k=1}^{m}{b}_{k}\left(X\left(t-\right)\right)d{Z}_{k}\left(t\right),\) $dX\left(t\right)=a\left(X,\left(t\right)\right)dt+{\sum }_{k=1}^m{b}_k\left(X,\left(t,-\right)\right)d{Z}_k\left(t\right),$

where Z_k, k ∈ {1, 2, . . . ,m}, are independent centered Lévy processes with finite second moments. We prove that if the coefficient a(x) has a certain power asymptotics as x ⟶ ∞ and the coefficients b_k, k ∈ {1, 2, . . . ,m}, satisfy certain growth condition, then the solution X(t) has the same asymptotics as the solution of the ordinary differential equation dx(t) = a(x(t))dt as t ⟶ ∞ a.s.