Coupling for Ornstein-Uhlenbeck processes with jumps

Abstract

Consider the linear stochastic differential equation (SDE) on Rn: dX t = AXt dt + B dLt, where A is a real n × n matrix, B is a real n × d real matrix and Lt is a Lévy process with Lévy measure ν on Rd. Assume that ν(dz) ≥ ρ0(z) dz for some ρ0 ≥ 0. If A ≤ 0, Rank(B) = n and ∫{|z-z0|≤} ρ0(z)-1 dz < ∞ holds for some z0 ε Rd and some ε > 0, then the associated Markov transition probability Pt (x, dy) satisfies ||Pt (x, ·)-Pt (y, ·)||var ≤ C(1 + |x - y|)/√t, x,y ε Rd, 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. © 2011 ISI/BS.