A Unified Approach to Coupling SDEs driven by L\'{e}vy Noise and Some Applications

We present a general method to construct couplings of stochastic differential equations driven by L'{e}vy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive L'{e}vy noise.