Optimal Periodic Double-Barrier Strategies for Spectrally Negative L\'{e}vy Processes

TL;DR

This paper analyzes an optimal control problem for spectrally negative Lévy processes, demonstrating the optimality of a double-barrier strategy with semi-explicit solutions and numerical validation.

Contribution

It introduces and proves the optimality of a double-barrier control strategy for spectrally negative Lévy processes, with semi-explicit solutions using scale functions.

Findings

Double-barrier strategy is optimal for the control problem.

Optimal strategy and value function are expressed semi-explicitly.

Numerical results validate the theoretical findings.

Abstract

We study a stochastic control problem where the underlying process follows a spectrally negative L\'{e}vy process. A controller can continuously increase the process but only decrease it at independent Poisson arrival times. We show the optimality of the double-barrier strategy, which increases the process whenever it would fall below some lower barrier and decreases it whenever it is observed above a higher barrier. An optimal strategy and the associated value function are written semi-explicitly using scale functions. Numerical results are also given.