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Stochastic optimal control of Lévy tax processes with bailouts

Dalal Al Ghanim, Ronnie Loeffen, Alexander R. Watson

TL;DR

The paper addresses optimal taxation and mandatory bailouts for a company whose capital evolves as a spectrally negative Lévy process. It develops a two-dimensional control framework where taxes are charged at a threshold-dependent rate when the controlled process hits new maxima and bailouts keep the process nonnegative with minimal cost, and proves that a threshold tax policy with minimal bailouts is optimal under a finite jump moment and bailout-cost condition. The analysis expresses the value function in terms of Lévy scale functions $W^{(q)}$ and $Z^{(q)}$, introduces the tax-reflection transform to construct natural tax processes, and provides a verification-based proof of optimality along with an explicit characterization of the optimal threshold and value function. A numerical example illustrates the computation of the optimal threshold and value, confirming the approach and highlighting how the optimal policy behaves with respect to model parameters. Overall, the work extends results from perturbed Brownian settings to general spectrally negative Lévy processes and offers a rigorous framework for optimal taxation with bailouts using natural tax processes and scale-function techniques.

Abstract

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.

Stochastic optimal control of Lévy tax processes with bailouts

TL;DR

The paper addresses optimal taxation and mandatory bailouts for a company whose capital evolves as a spectrally negative Lévy process. It develops a two-dimensional control framework where taxes are charged at a threshold-dependent rate when the controlled process hits new maxima and bailouts keep the process nonnegative with minimal cost, and proves that a threshold tax policy with minimal bailouts is optimal under a finite jump moment and bailout-cost condition. The analysis expresses the value function in terms of Lévy scale functions and , introduces the tax-reflection transform to construct natural tax processes, and provides a verification-based proof of optimality along with an explicit characterization of the optimal threshold and value function. A numerical example illustrates the computation of the optimal threshold and value, confirming the approach and highlighting how the optimal policy behaves with respect to model parameters. Overall, the work extends results from perturbed Brownian settings to general spectrally negative Lévy processes and offers a rigorous framework for optimal taxation with bailouts using natural tax processes and scale-function techniques.

Abstract

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.
Paper Structure (13 sections, 8 theorems, 72 equations, 3 figures)

This paper contains 13 sections, 8 theorems, 72 equations, 3 figures.

Table of Contents

  1. Key words and phrases.
  2. Introduction
  3. Natural tax processes with minimal bailouts
  4. Preliminaries on spectrally negative Lévy processes
  5. Definition and functionals of natural tax processes with minimal bailouts
  6. Solution of the optimal control problem
  7. Main result
  8. Example
  9. Proofs
  10. The tax-reflection transform
  11. Construction of the natural tax process with bailouts
  12. Verification lemma and proof of \ref{['p:value']}
  13. Proof of \ref{['t:control']}

Key Result

Proposition 1

Assume that $\int_1^\infty \theta\,\nu({\operator@font d} \theta) < \infty$. Let $(V,K)$ be the natural tax process with minimal bailouts with tax rate given by the threshold tax rate $\delta_b$ with threshold level $b\geq 0$, which is defined by Then for $(x,\bar{x})\in\mathbb R\times[0,\infty)$ such that $x\leq \bar{x}$ and for $\eta\geq 0$, the control $\pi = (\delta_b(V),K)$ is admissible and

Figures (3)

  • Figure 1: Illustration of the natural tax process $V$ with threshold tax rate at $b$ and minimal bailouts, where $\alpha = 0.3$, $\beta = 0.7$ and $b=2$. The blue (dashed) line is the path of the background Lévy process $X$. The red line is the process $V$. Note that at times when $V$ experiences a bailout, its instantaneous negative value is retained and the process is sent to the value zero immediately after.
  • Figure 2: Effect of tax rate bounds on the optimal threshold $b^*$ and the optimal value function at zero $v^*(0,0)$. The value of $\alpha$ has no effect on $b^*$.
  • Figure 3: Effect of varying $\eta$ on the optimal threshold $b^*$ and the optimal value function at zero $v^*(0,0)$, and on the value function $v^*(x,x)$.

Theorems & Definitions (19)

  • Definition 1
  • Proposition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Definition 2
  • ...and 9 more