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Let $A$ be a pseudo-differential operator with symbol $q(x,\xi)$. In thispaper we derive sufficient conditions which ensure the existence of a solutionto the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem. If the symbol $q$depends continuously on the space variable $x$, then the existence of solutionsis well understood, and therefore the focus lies on martingale problems forpseudo-differential operators with discontinuous coefficients. We prove anexistence result which allows us, in particular, to obtain new insights on theexistence of weak solutions to a class of L\'evy-driven SDEs with Borelmeasurable coefficients and on the the existence of stable-like processes withdiscontinuous coefficients. Moreover, we establish a Markovian selectiontheorem which shows that - under mild assumptions - the$(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem gives rise to a strongMarkov process. The result applies, in particular, to L\'evy-driven SDEs. Weillustrate the Markovian selection theorem with applications in the theory ofnon-local operators and equations; in particular, we establish under weakregularity assumptions a Harnack inequality for non-local operators of variableorder.

Existence of (Markovian) solutions to martingale problems associated with L\'evy-type operators