Coupling for Ornstein--Uhlenbeck processes with jumps

Details Coupling for Ornstein--Uhlenbeck processes with jumps Coupling for Ornstein--Uhlenbeck processes with jumps

2010-02-15

arxiv.org/abs/1002.2890v6

Bernoulli 2011, Vol. 17, No. 4, 1136-1158

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/10-BEJ308 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ308

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 L\'{e}vy process with L\'{e}vy 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 $A\le 0,\operatorname {Rank}(B)=n$ and $\int_{\{|z-z_0|\le\varepsilon\}}\rho_0(z)^{-1}\,\mathrm{d}z<\infty$ holds for some $z_0\in \mathbb{R}^d$ and some $\varepsilon>0$, 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.

Affiliated with

Also associated with

No associations

Rating

Coupling for Ornstein--Uhlenbeck processes with jumps does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Coupling for Ornstein--Uhlenbeck processes with jumps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Coupling for Ornstein--Uhlenbeck processes with jumps will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-146195