Infinite horizon impulse control of stochastic functional differential equations driven by Lévy processes

We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional Lp-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.