arXiv 1002.28902010-02math.PRmath.STstat.TH

带跳Ornstein-Uhlenbeck过程的耦合

Coupling for Ornstein--Uhlenbeck processes with jumps

Feng-Yu Wang

备注：: 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)
发表信息：: Bernoulli 2011, Vol. 17, No. 4, 1136-1158
DOI：: 10.3150/10-BEJ308

摘要：

考虑$\mathbb{R}^n$上的线性随机微分方程：\[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\]其中$A$是实$n\times n$矩阵，$B$是实$n\times d$矩阵，$L_t$是$\mathbb{R}^d$上具有Lévy测度$ν$的Lévy过程。假设对于某些$ρ_0\ge 0$，$ν(\mathrm {d}{z})\ge ρ_0(z)\,\mathrm{d}z$。如果$A\le 0,\operatorname {Rank}(B)=n$和$\int_{\{|z-z_0|\le\varepsilon\}}ρ_0(z)^{-1}\,\mathrm{d}z<\infty$对某些$z_0\in \mathbb{R}^d$和某些$\varepsilon>0$成立，则相关的马尔可夫转移概率$P_t(x,\mathrm {d}{y})$对某些常数$C>0$满足\[\|P_t(x,\cdot)-P_t(y,\cdot)\|_{\mathrm{var}}\le \frac{C(1+|x-y|)}{\sqrt{t}}, x,y\in \mathbb{R}^d,t>0,\]，这对于大的$t$是尖锐的，并且意味着过程具有成功的耦合。讨论了（条件）转移半群的Harnack不等式、超压缩性和强Feller性质。

英文摘要：

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 $ν$ on $\mathbb{R}^d$. Assume that $ν(\mathrm {d}{z})\ge ρ_0(z)\,\mathrm{d}z$ for some $ρ_0\ge 0$. If $A\le 0,\operatorname {Rank}(B)=n$ and $\int_{\{|z-z_0|\le\varepsilon\}}ρ_0(z)^{-1}\,\mathrm{d}z<\infty$ 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.

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英文摘要：

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