Refraction strategies in stochastic control: optimality for a general L\'evy process model

TL;DR

This paper proves the optimality of refraction strategies in a stochastic control problem driven by a general Le9vy process, extending previous results from spectrally negative cases to more general settings.

Contribution

It generalizes the optimality of refraction strategies to a broad class of Le9vy processes, beyond the spectrally negative case, under convex cost functions.

Findings

Refraction strategies are optimal for a wide class of Le9vy processes.

The result extends previous spectrally negative process cases.

Optimal control involves adjusting the drift at a constant rate when thresholds are exceeded.

Abstract

We revisit an absolutely-continuous version of the stochastic control problem driven by a L\'evy process. A strategy must be absolutely continuous with respect to the Lebesgue measure and the running cost function is assumed to be convex. We show the optimality of a refraction strategy, which adjusts the drift of the state process at a constant rate whenever it surpasses a certain threshold. The optimality holds for a general L\'evy process, generalizing the spectrally negative case presented in Hern\'andez-Hern\'andez et al.(2016).