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Optimal stopping of the stable process with state-dependent killing

K. van Schaik, A. R. Watson, X. Xu

Abstract

We describe the solution of an optimal stopping problem for a stable Lévy process killed at state-dependent rate, which can be seen as a model for bankruptcy. The killing rate is chosen in such a way that the killed process remains self-similar, and the solution to the optimal stopping problem is obtained by characterising a self-similar Markov process associated with the stable process. The optimal stopping strategy is to stop upon first passage into an interval, found explicitly in terms of the parameters of the model.

Optimal stopping of the stable process with state-dependent killing

Abstract

We describe the solution of an optimal stopping problem for a stable Lévy process killed at state-dependent rate, which can be seen as a model for bankruptcy. The killing rate is chosen in such a way that the killed process remains self-similar, and the solution to the optimal stopping problem is obtained by characterising a self-similar Markov process associated with the stable process. The optimal stopping strategy is to stop upon first passage into an interval, found explicitly in terms of the parameters of the model.
Paper Structure (8 sections, 14 theorems, 69 equations)

This paper contains 8 sections, 14 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. The killed path-censored stable process and its Lamperti transform
  3. An optimal stopping problem for the Lamperti transform xi
  4. Identification of the Lamperti transform xi
  5. The optimal stopping problem as a perpetual call option
  6. Solution of the optimal stopping problem
  7. Remark on a variant stopping problem
  8. Acknowledgments

Key Result

Theorem 1

There exists $\delta > \max(0,\alpha-1)$, uniquely characterised in terms of the parameters of the stable process and the killing coefficient $k$, such that the following holds for the process $X$ started in $x \ne 0$.

Theorems & Definitions (29)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Killing probability
  • proof
  • Proposition 2: Structure of $\xi$
  • ...and 19 more