Random time changes of Feller processes

Abstract

We show that the SDE $d{X}_t=\sigma (X_{t-})d{L}_t$, $X_0\sim \mu$ driven by a one-dimensional symnmetric $\alpha$-stable Lévy process $(L_t{)}_{t\ge 0}$, $\alpha \in (0,2]$, has a unique weak solution for any continuous function $\sigma :\mathbb{R}\to (0,\infty )$ which grows at most linearly. Our approach relies on random time changes of Feller processes. We study under which assumptions the random-time change of a Feller process is a conservative $C_b$-Feller process and prove the existence of a class of Feller processes with decomposable symbols. In particular, we establish new existence results for Feller processes with unbounded coefficients. As a by-product, we obtain a sufficient condition in terms of the symbol of a Feller process $(X_t{)}_{t\ge 0}$ for the perpetual integral ${\int }_{(0,\infty )}f(X_s)ds$ to be infinite almost surely.

References in corpus (1)

• On Martingale Problems and Feller Processes

Cited by in corpus (4)

• Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear Lévy(-type) processes
• Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators
• Schauder estimates for Poisson equations associated with non-local Feller generators
• On the explosion of the number of fragments in the simple exchangeable fragmentation-coalescence processes