Moment bounds for dissipative semimartingales with heavy jumps

In this paper we show that if large jumps of an It\^o-semimartingale $X$ have a finite $p$-moment, $p>0$, the radial part of its drift is dominated by $-|X|^\kappa$ for some $\kappa\geq -1$, and the balance condition $p+\kappa>1$ holds true, then under some further natural technical assumptions $\sup_{t\geq 0} \mathbf{E} |X_t|^{p_X}<\infty$ for each $p_X\in(0,p+\kappa-1)$. The upper bound $p+\kappa-1$ is generically optimal. The proof is based on the extension of the method of Lyapunov functions to the semimartingale framework. The uniform moment estimates obtained in this paper are indispensable for the analysis of ergodic properties of L\'evy driven stochastic differential equations and L\'evy driven multi-scale systems.