Semigroup properties of solutions of SDEs driven by L{é}vy processes with independent coordinates

Abstract

We study the stochastic differential equation $d{X}_t=A(X_{t-})d{Z}_t$, $X_0=x$, where $Z_t=(Z_t^{(1)},\dots ,Z_t^{(d)}{)}^T$ and $Z_t^{(1)},\dots ,Z_t^{(d)}$ are independent one-dimensional L{é}vy processes with characteristic exponents ${\psi }_1,\dots ,{\psi }_d$. We assume that each ${\psi }_i$ satisfies a weak lower scaling condition WLSC($\alpha ,0,\underline{C}$), a weak upper scaling condition WUSC($\beta ,1,\overline{C}$) (where $0<\alpha \le \beta <2$) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all ${\psi }_1,\dots ,{\psi }_d$ are the same and $\alpha ,\beta$ are arbitrary, or (ii) not all ${\psi }_1,\dots ,{\psi }_d$ are the same and $\alpha >(2/3)\beta$. We also assume that the determinant of $A(x)=(a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed $\gamma \in (0,\alpha )\cap (0,1]$ the semigroup $P_t$ of the process $X$ satisfies $|P_{t}f(x)-P_{t}f(y)|\le c{t}^{-\gamma /\alpha }|x-y{|}^{\gamma }||f|{|}_{\infty }$ for arbitrary bounded Borel function $f$. We also show the existence of a transition density of the process $X$.

arXiv admin note: substantial text overlap with arXiv:1811.05960