Constructions of coupling processes for Lévy processes

Abstract

We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.

Harnack inequalities for stochastic equations driven by Lévy noise

“…(3) The gradient inequality in Theorem 2.1(2) generalizes the main results in [16,3,12] for Lévy processes or linear equations driven by Lévy noise. When K ≤ 0 and λ is bounded, it follows from Theorem 2.1(2) that…”

“…In fact, in this case we may split L t into two independent Lévy parts, where one of them has Lévy measure c|x| −d B(|x| −2 )dx and is thus a subordinate Brownian motion (cf. [16,3]), and the integral of σ w.r.t. the other can be combined with the term V t .…”

Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes

“…Previous attempts at constructing couplings of Lévy processes or couplings of solutions to Lévy-driven SDEs include e.g. a coupling of subordinate Brownian motions by making use of the coupling of Brownian motions by reflection (see [2]), a coupling of compound Poisson processes obtained from certain couplings of random walks (see [22] for the original construction and [31] for a related idea applied to Lévy-driven SDEs) and a combination of the coupling by reflection and the synchronous coupling defined via its generator for solutions to SDEs driven by Lévy processes with a symmetric α-stable component (see [32]). In the present paper we use a different idea for a coupling, as well as a different method of construction.…”

Coupling and Applications

“…For detailed proofs of the above results and further developments on couplings and applications of Lévy processes, one may check with recent papers [5,22,23,32,33].…”