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Moment bounds for dissipative semimartingales with heavy jumps

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  • Kulik, Alexei
  • Pavlyukevich, Ilya

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.

Suggested Citation

  • Kulik, Alexei & Pavlyukevich, Ilya, 2021. "Moment bounds for dissipative semimartingales with heavy jumps," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 274-308.
    • Handle: RePEc:eee:spapps:v:141:y:2021:i:c:p:274-308
    DOI: 10.1016/j.spa.2021.07.004
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    References listed on IDEAS

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    1. Samorodnitsky, G. & Grigoriu, M., 2003. "Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Lévy motions," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 69-97, May.
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    3. Arturo Kohatsu & Makoto Yamazato, 2003. "On moments and tail behaviors of storage processes," Economics Working Papers 673, Department of Economics and Business, Universitat Pompeu Fabra.
    4. Douc, Randal & Fort, Gersende & Guillin, Arnaud, 2009. "Subgeometric rates of convergence of f-ergodic strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 897-923, March.
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