For spectrally negative Lévy processes, the optimal policy with continuous upward and Poisson-timed downward control is a double-barrier strategy with a semi-explicit value function.
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Optimal Periodic Double-Barrier Strategies for Spectrally Negative L\'{e}vy Processes
For spectrally negative Lévy processes, the optimal policy with continuous upward and Poisson-timed downward control is a double-barrier strategy with a semi-explicit value function.