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

Author: Franziska Kühn
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 60J35; Secondary 60H10, 60G51, 60J25, 60J75, 60G44
Published electronically: February 21, 2018
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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)

$\displaystyle dX_t = \sigma (X_{t-}) \, dL_t, \qquad X_0 \sim \mu ,$    

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 Lévy measure $ \nu $ of the driving Lévy process $ (L_t)_{t \geq 0}$ satisfies

$\displaystyle \nu (\{y \in \mathbb{R}^k; \vert\sigma (x)y+x\vert<r\}) \xrightarrow []{\vert x\vert \to \infty } 0.$    

This generalizes a result by Schilling & Schnurr (2010) which states that the solution to the SDE has this property if $ \sigma $ is bounded.

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Additional Information

Franziska Kühn
Affiliation: Institut für Mathematische Stochastik, Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germany

Keywords: Feller process, stochastic differential equation, unbounded coefficients.
Received by editor(s): October 7, 2016
Received by editor(s) in revised form: November 17, 2016, May 10, 2017, and October 23, 2017
Published electronically: February 21, 2018
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2018 American Mathematical Society

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