Solutions of L\'evy-driven SDEs with unbounded coefficients as Feller processes

TL;DR

This paper establishes conditions under which solutions to certain Lévy-driven SDEs with unbounded coefficients are Feller processes, linking the behavior of the Lévy measure at infinity to the process's generator domain.

Contribution

It provides a characterization of when solutions to Lévy-driven SDEs with unbounded coefficients are Feller processes based on the Lévy measure's properties at infinity.

Findings

Solution is a Feller process under specified Lévy measure conditions.

Domain of the generator includes smooth compactly supported functions.

Characterization of Feller property linked to Lévy measure decay at infinity.

Abstract

Let $(L_t)_{t \geq 0}$ be a $k$-dimensional L\'evy process and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times k}$ a continuous function such that the L\'evy-driven stochastic differential equation (SDE) $$dX_t = \sigma(X_{t-}) \, dL_t, \qquad X_0 \sim \mu$$ 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 L\'evy measure $\nu$ of the driving L\'evy process $(L_t)_{t \geq 0}$ satisfies $$\nu(\{y \in \mathbb{R}^k; |\sigma(x)y+x|<r\}) \xrightarrow[]{|x| \to \infty} 0.$$