$L^p$-Wasserstein distance for stochastic differential equations driven by Lévy processes

Abstract

Coupling by reflection mixed with synchronous coupling is constructed for a class of stochastic differential equations (SDEs) driven by Lévy noises. As an application, we establish the exponential contractivity of the associated semigroups $(P_t{)}_{t\ge 0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in [1,\infty )$. In particular, consider the following SDE: \[\mathrm{d}X_t=\mathrm{d}Z_t+b(X_t)\,\mathrm{d}t,\] where $(Z_t{)}_{t\ge 0}$ is a symmetric $\alpha$-stable process on ${\mathbb{R}}^d$ with $\alpha \in (1,2)$. We show that if the drift term $b$ satisfies that for any $x,y\in {\mathbb{R}}^d$, \[\bigl\langle b(x)-b(y),x-y\bigr\rangle\le\cases{K_1|x-y|^2,\qquad |x-y|\le L_0;\cr -K_2|x-y|^θ,\qquad |x-y|>L_0}\] holds with some positive constants $K_1$, $K_2$, $L_0>0$ and $\theta \ge 2$, then there is a constant $\lambda :=\lambda (\theta ,K_1,K_2,L_0)>0$ such that for all $p\in [1,\infty )$, $t>0$ and $x,y\in {\mathbb{R}}^d$, \[W_p(δ_xP_t,δ_yP_t)\le C(p,θ,K_1,K_2,L_0)\mathrm{e}^{-λt/p}\biggl[\frac{|x-y|^{1/p}\vee|x-y|}{1+|x-y|{\mathbf{1}}_{(1,\infty )\times (2,\infty)}(t,θ)}\biggr].\]

Published at http://dx.doi.org/10.3150/15-BEJ705 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (1)

• Exponential Convergence in $L^p$-Wasserstein Distance for Diffusion Processes without Uniformly Dissipative Drift

Cited by in corpus (4)

• Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes
• Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling
• A Unified Approach to Coupling SDEs driven by Lévy Noise and Some Applications
• On Sub-Geometric Ergodicity of Diffusion Processes