Optimal Periodic Double-Barrier Strategies for Spectrally Negative Lévy Processes

TL;DR

This work analyzes a stochastic control problem for a state following a spectrally negative Lévy process, where upward control is continuous but downward control occurs only at Poisson observation times. It proves the optimality of a periodic-classical double-barrier strategy $(a^*,b^*)$, and expresses the associated value function semi-explicitly using scale functions $W^{(q)}$, $Z^{(q)}$, and related two-parameter objects, while establishing existence, uniqueness, and verification of the barriers. The results extend continuous two-sided control to discrete observation opportunities, provide semi-explicit fluctuation-theoretic formulas, and are supported by numerical experiments illustrating barrier convergence to the classical solution as the observation rate increases and the single-barrier regime when the running-cost slope condition fails. Overall, the findings offer tractable, provably optimal policies for inventory, finance, and insurance settings with constrained downward control and random observation opportunities.

Abstract

We study a stochastic control problem where the underlying process follows a spectrally negative Lévy 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.

Figures (3)

• Figure 1: Left: Sample path of a spectrally negative Lévy process $X$. Right: Corresponding controlled process $Y^{a, b}$ with $a = -6$ and $b = 0$. Vertical dashed lines indicate the Poisson arrival times $\mathcal{T}_r$. Among the three arrival times displayed, $Y^{a, b}$ exceeds $b = 0$ only at the second one, causing downward control to activate there but not at the first or third arrival times.
• Figure 2: Plots of $v_{a, b}$ (for $r = 0.1$) against initial position $x$. Left: Plot of $v_{a^*, b}$ (blue dashed) for $b = b^* - 5, b^* - 4, \dots, b^* - 1, b^* + 1, \dots, b^* + 5$, with $(a^*, v_{a^*, b}(a^*))$ (lime triangle), $(a^*, v_{a^*, b^*}(a^*))$ (red triangle), $(b, v_{a^*, b}(b))$ (lime square), and $(b^*, v_{a^*, b^*}(b^*))$ (red square). Right: Plot of $v_{a, b^*}$ (blue dashed) for $a = a^* - 5, a^* - 4, \dots, a^* - 1, a^* + 1, \dots, a^* + 5$, with $(a, v_{a, b^*}(a))$ (lime triangle), $(a^*, v_{a^*, b^*}(a^*))$ (red triangle), $(b^*, v_{a, b^*}(b^*))$ (lime square), and $(b^*, v_{a^*, b^*}(b^*))$ (red square). In both plots, the value function $v_{a^*, b^*}$ is indicated by red curves.
• Figure 3: Left: Plot of periodic-classical barriers $(a^*, b^*)$ for $r = 0.1, 0.2, \dots, 0.9, 1, 2, \dots, 9, 10, 20, \newline\dots, 90, 100, 200, \dots, 900$ (squares) and classical barriers (red dashed lines). Right: Plot of $v_{a^*, b^*}$ (blue dashed) for $r = 0.1, 0.2, \dots, 0.9, 1, 2, \dots, 9, 10, 20, \dots, 90$, with $(a^*, v_{a^*, b^*}(a^*))$ (lime triangles) and $(b^*, v_{a^*, b^*}(b^*))$ (lime squares), alongside the classical value function (red solid line), with lower and upper barriers marked by a red triangle and square, respectively.

Theorems & Definitions (37)

• Remark 2.1
• Remark 3.1
• Lemma 3.2: Lemma 3.1 in noba_optimal_2018
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• proof
• Lemma 3.7
• Proposition 3.8