## Abstract

Under the assumption that the asset value follows a phase-type jump-diffusion, we show that the expected discounted penalty satisfies an ODE and obtain a general form for the expected discounted penalty. In particular, if only downward jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if the downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Lévy measure), we obtain closed-form solutions for the expected discounted penalty. As an application, we work out an example in Leland’s structural model with jumps. For earlier and related results, see Gerber and Landry et al. (1998), Hilberink and Rogers et al. (2002), Asmussen et al. (2004), and Kyprianou and Surya et al. (2007).

Original language | English |
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Title of host publication | Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning (In 4 Volumes) |

Publisher | World Scientific Publishing Co. |

Pages | 1561-1598 |

Number of pages | 38 |

ISBN (Electronic) | 9789811202391 |

ISBN (Print) | 9789811202384 |

DOIs | |

State | Published - 1 Jan 2020 |

## Keywords

- Expected discounted penalty
- Jump-diffusion
- Optimal capital structure
- Phase-type distribution