Moment bounds for dissipative semimartingales with heavy jumps

Alexei Kulik and Ilya Pavlyukevich

Stochastic Processes and their Applications, 2021, vol. 141, issue C, 274-308

Abstract: In this paper we show that if large jumps of an Itô-semimartingale X have a finite p-moment, p>0, the radial part of its drift is dominated by −|X|κ for some κ≥−1, and the balance condition p+κ>1 holds true, then under some further natural technical assumptions one has supt≥0E|Xt|pX<∞ for each pX∈(0,p+κ−1). The upper bound p+κ−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 Lévy driven stochastic differential equations and Lévy driven multi-scale systems.

Keywords: Long-time moment bounds; Lyapunov function; Cesàro mean; Heavy tails; Dissipative system; Passage times (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1016/j.spa.2021.07.004

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