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Semigroup properties of solutions of SDEs driven by Lévy processes with independent coordinates

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  • Kulczycki, Tadeusz
  • Ryznar, Michał

Abstract

We study the stochastic differential equation dXt=A(Xt−)dZt, X0=x, where Zt=(Zt(1),…,Zt(d))T and Zt(1),…,Zt(d) are independent one-dimensional Lévy processes with characteristic exponents ψ1,…,ψd. We assume that each ψi satisfies a weak lower scaling condition WLSC(α,0,C̲), a weak upper scaling condition WUSC(β,1,C¯) (where 0<α≤β<2) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all ψ1,…,ψd are the same and α,β are arbitrary, or (ii) not all ψ1,…,ψd are the same and α>(2∕3)β. We also assume that the determinant of A(x)=(aij(x)) is bounded away from zero, and aij(x) are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed γ∈(0,α)∩(0,1] the semigroup Pt of the process X satisfies |Ptf(x)−Ptf(y)|≤ct−γ∕α|x−y|γ||f||∞ for arbitrary bounded Borel function f. We also show the existence of a transition density of the process X.

Suggested Citation

  • Kulczycki, Tadeusz & Ryznar, Michał, 2020. "Semigroup properties of solutions of SDEs driven by Lévy processes with independent coordinates," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7185-7217.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:12:p:7185-7217
    DOI: 10.1016/j.spa.2020.07.011
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    References listed on IDEAS

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    1. Paweł Sztonyk, 2017. "Estimates of densities for Lévy processes with lower intensity of large jumps," Mathematische Nachrichten, Wiley Blackwell, vol. 290(1), pages 120-141, January.
    2. Liang, Mingjie & Wang, Jian, 2020. "Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3053-3094.
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    Cited by:

    1. Oleksii Kulyk, 2023. "Support Theorem for Lévy-driven Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1720-1742, September.

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