Stochastic optimal control of Lévy tax processes with bailouts

We consider controlling the paths of a spectrally negative Lévy 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 Lévy processes to which the optimal solution belongs, which extends a known result for perturbed Brownian motions to the case of a general Lévy process with no positive jumps.

💡 Research Summary

This paper studies a stochastic optimal control problem for a spectrally negative Lévy process that models the capital of an insurance company. Two control mechanisms are available: (i) a tax that is levied only when the controlled capital process attains a new running maximum, and (ii) mandatory capital injections (bailouts) that keep the capital non‑negative. The tax rate is constrained to lie between a lower bound α ≥ 0 and an upper bound β ∈ (0,1), while the bailout cost is multiplied by a penalty factor η ≥ 1. The objective is to maximise the expected discounted net revenue, i.e. the present value of tax receipts minus the present value of bailout costs.

The authors formulate the admissible control pair (H,L) where Hₜ represents the instantaneous tax rate and Lₜ the cumulative amount of bailouts. The controlled capital process is defined as
Uₜ = Xₜ + Lₜ − ∫₀ᵗ Hₛ d(X+L)ₛ,
where X is the underlying spectrally negative Lévy process. Taxes are only paid when Uₜ coincides with its running supremum, a property that follows from Lemma 10. The optimisation problem is to find
v⁎(x, \bar{x}) = sup_{(H,L)∈Π_{x,\bar{x}}} E_{x,\bar{x}}\Big