Stochastic optimal control of L\'evy tax processes with bailouts

TL;DR

This paper develops a new two-dimensional stochastic control model for spectrally negative Le9vy processes involving taxes and bailouts, extending existing models to more general processes and providing explicit optimal strategies.

Contribution

It introduces a larger class of taxation controls and characterizes optimal strategies for a broad class of controlled Le9vy processes, extending previous results beyond Brownian motions.

Findings

Derived explicit optimal control strategies for the model.

Extended the class of controlled Le9vy processes beyond Brownian cases.

Provided a comprehensive characterization of the optimal solutions.

Abstract

We consider controlling the paths of a spectrally negative L\'evy process by two means: the subtraction of `taxes' when the process is at an all-time maximum, and the addition of `bailouts' which keep the value of the process above zero. We solve the corresponding stochastic optimal control problem of maximising the expected present value of the difference between taxes received and cost of bailouts given. Our class of taxation controls is larger than has been considered up till now in the literature and makes the problem truly two-dimensional rather than one-dimensional. Along the way, we define and characterise a large class of controlled L\'evy processes to which the optimal solution belongs, which extends a known result for perturbed Brownian motions to the case of a general L\'evy process with no positive jumps.