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

Abstract

Let $(L_t{)}_{t\ge 0}$ 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) $$d{X}_t=\sigma (X_{t-})d{L}_t,X_0\sim \mu$$ 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 Lévy measure $\nu$ of the driving Lévy process $(L_t{)}_{t\ge 0}$ satisfies $$\nu (\{y\in {\mathbb{R}}^k;|\sigma (x)y+x|<r\})\stackrel{|x|\to \infty }{\to }0.$$

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• On the domain of fractional Laplacians and related generators of Feller processes
• Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators
• Transition probabilities of Lévy-type processes: Parametrix construction
• Multifractal properties of sample paths of ground state-transformed jump processes
• Schauder estimates for Poisson equations associated with non-local Feller generators
• Short-time behavior of solutions to Lévy-driven SDEs
• Kolmogorov Modes and Linear Response of Jump-Diffusion Models