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A potential-theoretic approach to optimal stopping in a spectrally Lévy Model

Masahiko Egami, Tomohiro Koike

Abstract

We establish a systematic solution method for optimal stopping problems of spectrally negative Lévy processes. Our approach relies essentially on the potential theory, in particular the Riesz decomposition and the maximum principle. Using these mathematical results, we not only derive necessary and sufficient conditions of optimality for a broad class of reward functions, but also develop a method to tackle general problems in a direct and constructive way (without pre-specifying the solution form). To reinforce the latter point, we provide a step-by-step solution procedure applicable to complex solution structures, including continuation regions with multiple connected components.

A potential-theoretic approach to optimal stopping in a spectrally Lévy Model

Abstract

We establish a systematic solution method for optimal stopping problems of spectrally negative Lévy processes. Our approach relies essentially on the potential theory, in particular the Riesz decomposition and the maximum principle. Using these mathematical results, we not only derive necessary and sufficient conditions of optimality for a broad class of reward functions, but also develop a method to tackle general problems in a direct and constructive way (without pre-specifying the solution form). To reinforce the latter point, we provide a step-by-step solution procedure applicable to complex solution structures, including continuation regions with multiple connected components.

Paper Structure

This paper contains 20 sections, 23 theorems, 69 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The problem
  3. Literature review and research motivation
  4. Mathematical tools
  5. Spectrally negative Lévy processes and their scale functions
  6. The potential theory
  7. The smooth Gerber–Shiu function
  8. The maximum principle
  9. The Riesz representation of $H_{A^c}^q g$
  10. Necessary conditions
  11. Sufficient conditions
  12. The case $n=1$
  13. The case $n=2$
  14. General procedure to solve optimal stopping
  15. The case where the optimal boundary lies on the irregular set $F$
  16. ...and 5 more sections

Key Result

Proposition 2.1

A function $u$ is a $q$-potential if and only if there exists a measure $\mu$ such that $u = G_q\mu := \int_{(-\infty, \infty)} G_q(\cdot, y) \mu({\rm d} y).$

Figures (9)

  • Figure 1: The real line with the points $a_0 ,\ell_0$ and the subharmonic components. The red segments represent the set $D =(\ell_1, r_1)\cup (\ell_2, r_2) \cup \{\ell_3\} \cup (\ell_0, \infty)$, which consists of four subharmonic components. The blue segment represents $(-\infty, a)$ where $g$ is superharmonic.
  • Figure 2: Relative position of $h_a$ and $g$ is presented. Though $h_{a'}$ is not majorant of $g$, $h_a$ is a majorant of $g$ if we take $a$ sufficiently large in the negative direction. Hence, $g$ in this figure satisfies Condition \ref{['conditionA']}.
  • Figure 3: $g$ is shown as a black curve, and $g_1$ is shown as a blue curve. The figure illustrates that the smooth Gerber–Shiu function $h_{a'}$ for $g$ at $a'$ coincides with the smooth Gerber–Shiu function $h^1_{b'}$ for $g_1$ at $b'$ on the interval $(b', \infty)$.
  • Figure 4: Blue: $g_M(x) = (K - e^x)_+$, red: $v_M$; a single one-sided continuation region emerges.
  • Figure 5: Blue: $g(x) = (K - e^x)\mathbf{1}_{\{x \le d\}} + \max\!\{(K - e^d) - l(x - d),\, 0\}\mathbf{1}_{\{x \ge d\}},$ red: $v$; one- and two-sided continuation regions appear.
  • ...and 4 more figures

Theorems & Definitions (51)

  • Proposition 2.1
  • Proposition 2.2: Riesz decomposition
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • ...and 41 more