Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes

Abstract:

Let $(L_t)_{t \geq 0}$ be a $k$-dimensional Lévy process and $\sigma : \mathbb {R}^d \to \mathbb {R}^{d \times k}$ a continuous function such that the Lévy-driven stochastic differential equation (SDE) \begin{equation*} dX_t = \sigma (X_{t-}) dL_t, \qquad X_0 \sim \mu , \end{equation*} has a unique weak solution. We show that the solution is a Feller process whose domain of the generator contains the smooth functions with compact support if and only if the Lévy measure $\nu$ of the driving Lévy process $(L_t)_{t \geq 0}$ satisfies \begin{equation*} \nu (\{y \in \mathbb {R}^k; |\sigma (x)y+x|<r\}) \xrightarrow []{|x| \to \infty } 0. \end{equation*} This generalizes a result by Schilling & Schnurr (2010) which states that the solution to the SDE has this property if $\sigma$ is bounded.