Coupling for Ornstein–Uhlenbeck processes with jumps

Abstract

Consider the linear stochastic differential equation (SDE) on $ℝ^n$: $$\mathrm{d}X_t = AX_t \mathrm{d}t + B \mathrm{d}L_t,$$ where $A$ is a real $n × n$ matrix, $B$ is a real $n × d$ real matrix and $L_t$ is a Lévy process with Lévy measure $ν$ on $ℝ^d$. Assume that $ν(\mathrm{d}z) ≥ ρ_0(z)\mathrm{d}z$ for some ${ρ_0} \geq 0$. If $A \leq 0$, Rank$(B) = n$ and $∫_{\{|z−z_0| \leq ε\}}ρ_0(z)^{−1} \mathrm{d}z< \infty$ holds for some $z_0 ∈ ℝ^d$ and some $ε > 0$, then the associated Markov transition probability $P_t(x, \mathrm{d}y)$ satisfies $$‖P_t(x, ⋅) − P_t(y, ⋅)‖_{\mathrm{var}} \le \frac{C(1 + |x − y|)}{√t}, x, y ∈ ℝ^d, t > 0,$$ 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.