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

Tadeusz Kulczycki and Michał Ryznar

Stochastic Processes and their Applications, 2020, vol. 130, issue 12, 7185-7217

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.

Date: 2020
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DOI: 10.1016/j.spa.2020.07.011

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