On stochastic control under Poisson observations: optimality of a barrier strategy in a general L\'evy model

TL;DR

This paper investigates a stochastic control problem with Poisson observation times in a general Lévy process setting, demonstrating the optimality of a barrier strategy and its convergence to continuous observation solutions.

Contribution

It proves the optimality of a barrier strategy under Poisson observations in a general Lévy model, extending continuous observation results.

Findings

Barrier strategy is optimal under Poisson observations.

Optimal solutions converge to continuous observation solutions.

Results apply to a broad class of Lévy processes.

Abstract

We study a version of the stochastic control problem of minimizing the sum of running and controlling costs, where control opportunities are restricted to independent Poisson arrival times. Under a general setting driven by a general L\'evy process, we show the optimality of a periodic barrier strategy, which moves the process upward to the barrier whenever it is observed to be below it. The convergence of the optimal solutions to those in the continuous-observation case is also shown.