On Asymptotics of Solutions of Stochastic Differential Equations with Jumps

Abstract

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 $Z_k, \ k \in \{1, 2, ..., m\}$ are independent centered L\'evy processes with finite second moments. We prove that if coefficient $a(x)$ has certain power asymptotics as $x \to \infty$ and coefficients $b_k, \ k \in \{1, 2, ..., m\},$ 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 $t \to \infty$ a.s.