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

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We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional L p -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.