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We provide verification theorems (at different levels of generality) forinfinite horizon stochastic control problems in continuous time forsemimartingales. The control framework is given as an abstract "martingaleformulation", which encompasses a broad range of standard control problems.Under appropriate conditions we show that the set of admissible controls givesrise to a certain class of controlled special semimartingales. Our resultsgeneralise both the standard controlled It\^o- and L\'evy-diffusion settings aswe allow ourselves to locally control not only the drift and diffusioncoefficients, but also the jump intensity measure of the jumps. As anillustration, we present a few examples with explicit solutions.

Martingale approach to control for general jump processes