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Dynkin Games for Lévy Processes

Laura Aspirot, Ernesto Mordecki, Andres Sosa

Abstract

We obtain a verification theorem for solving a Dynkin game driven by a Lévy process. The result requires finding two averaging functions that, composed respectively with the supremum and the infimum of the process, summed, and taked the expectation, provide the value function of the game. The optimal stopping rules are the respective hitting times of the support sets of the averaging functions. The proof relies on fluctuation identities of the underlying Lévy process. We illustrate our result with three new simple examples, where the smooth pasting property of the solutions is not always present.

Dynkin Games for Lévy Processes

Abstract

We obtain a verification theorem for solving a Dynkin game driven by a Lévy process. The result requires finding two averaging functions that, composed respectively with the supremum and the infimum of the process, summed, and taked the expectation, provide the value function of the game. The optimal stopping rules are the respective hitting times of the support sets of the averaging functions. The proof relies on fluctuation identities of the underlying Lévy process. We illustrate our result with three new simple examples, where the smooth pasting property of the solutions is not always present.

Paper Structure

This paper contains 13 sections, 11 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Proof of Theorem \ref{['theorem:1']}
  5. Smooth pasting
  6. Integrability
  7. Three processes with affine payoff
  8. Brownian motion with drift
  9. Cramér-Lundberg process
  10. Compound Poisson process
  11. Callable futures under the Kou model
  12. Averaging functions and thresholds
  13. Conclusions

Key Result

Proposition 1

Consider a pair $(\tilde{\sigma},\tilde{\tau})$ of stopping times for the problem in Definition def:dg and a real valued function $\widetilde{V}(x)$, defined by Assume that the two following conditions hold Then, the DG in Definition def:dg has value function $\widetilde{V}(x)$, and $(\tilde{\sigma},\tilde{\tau})$ is a pair of optimal stopping times for the problem, meaning that eq:dg holds.

Figures (4)

  • Figure 1: Brownian Motion. Value function $V(x)$ (solid), $G_1(x)$, $G_2(x)$ (dashed) and $Q_I(x)$, $Q_S(x)$ (dotted). Left: symmetric case. Right: asymmetric case.
  • Figure 2: Cramér-Lundberg process. Value function $V(x)$ (solid), $G_1(x)$, $G_2(x)$ (dashed) and $Q_I(x)$, $Q_S(x)$ (dotted).
  • Figure 3: Compound Poisson process. Value function $V(x)$ (solid), $G_1(x)$, $G_2(x)$ (dashed) and $Q_I(x)$, $Q_S(x)$ (dotted). Left: symmetric case. Right: asymmetric case.
  • Figure 4: Callable future under the Kou model. Value function $V(x)$ (solid), $G_1(x)$, $G_2(x)$ (dashed) and $Q_I(x)$, $Q_S(x)$ (dotted).

Theorems & Definitions (20)

  • Definition 1: Dynkin game
  • Proposition 1
  • Theorem 1
  • Remark 1
  • Corollary 1
  • proof
  • Lemma 1: MG
  • proof
  • Lemma 2: SPMG-SBMG
  • proof
  • ...and 10 more