On the аsymptotics of solutions of stochastic differential equations with jumps
Abstract
UDC 519.21
Consider a one-dimensional stochastic differential equation with jumps $$dX(t) = a(X(t))dt + \sum_{k = 1}^m b_k(X(t-))dZ_k(t),$$ where $Z_k,$ $k \in \{1, 2, \ldots , m\},$ are independent centered L\'evy processes with finite second moments. We prove that if the coefficient $a(x)$ has a certain power asymptotics as $x \to \infty$ and the coefficients $b_k,$ $ k \in \{1, 2, \ldots , m\},$ satisfy certain growth condition, then a solution $X(t)$ has the same asymptotics as the solution of $d x(t) = a(x(t))d t$ as $t \to \infty$ a.s.
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