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By using the mirror coupling for solutions of SDEs driven by pure jump L\'evyprocesses, we extend some transportation and concentration inequalities, whichwere previously known only in the case where the coefficients in the equationsatisfy a global dissipativity condition. Furthermore, by using the mirrorcoupling for the jump part and the coupling by reflection for the Brownianpart, we extend analogous results for jump diffusions. To this end, we improvesome previous results concerning such couplings and show how to combine thejump and the Brownian case. As a crucial step in our proof, we develop a novelmethod of bounding Malliavin derivatives of solutions of SDEs with both jumpand Gaussian noise, which involves the coupling technique and which might be ofindependent interest. The bounds we obtain are new even in the case ofdiffusions without jumps.

Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling