Coupling for Ornstein--Uhlenbeck processes with jumps

Abstract

Consider the linear stochastic differential equation (SDE) on ${\mathbb{R}}^n$: \[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\] where $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ real matrix and $L_t$ is a Lévy process with Lévy measure $\nu$ on ${\mathbb{R}}^d$. Assume that $\nu (\mathrm{d}z)\ge {\rho }_0(z)\mathrm{d}z$ for some ${\rho }_0\ge 0$. If $A\le 0,\mathrm{Rank}(B)=n$ and ${\int }_{\{|z-z_0|\le \epsilon \}}{\rho }_0(z{)}^{-1}\mathrm{d}z<\infty$ holds for some $z_0\in {\mathbb{R}}^d$ and some $\epsilon >0$, then the associated Markov transition probability $P_t(x,\mathrm{d}y)$ satisfies \[\|P_t(x,\cdot)-P_t(y,\cdot)\|_{\mathrm{var}}\le \frac{C(1+|x-y|)}{\sqrt{t}}, x,y\in \mathbb{R}^d,t>0,\] for some constant $C>0$, which is sharp for large $t$ and implies that the process has successful couplings. The Harnack inequality, ultracontractivity and the strong Feller property are also investigated for the (conditional) transition semigroup.

Published in at http://dx.doi.org/10.3150/10-BEJ308 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Densities for Ornstein-Uhlenbeck processes with jumps
• Stochastic analysis on configuration spaces: basic ideas and recent results
• A note on the existence of transition probability densities for Lévy processes

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• Coupling and Strong Feller for Jump Processes on Banach Spaces
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• Jump Type Stochastic Differential Equations with Non-Lipschitz Coefficients: Non Confluence, Feller and Strong Feller Properties, and Exponential Ergodicity
• On a Cameron--Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion
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