TY - CHAP

T1 - An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model

AU - Chen, Yu Ting

AU - Lee, Cheng Few

AU - Sheu, Yuan Chung

N1 - Publisher Copyright:
© 2021 by World Scientific Publishing Co. Pte. Ltd.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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).

AB - 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).

KW - Expected discounted penalty

KW - Jump-diffusion

KW - Optimal capital structure

KW - Phase-type distribution

UR - http://www.scopus.com/inward/record.url?scp=85096281138&partnerID=8YFLogxK

U2 - 10.1142/9789811202391_0041

DO - 10.1142/9789811202391_0041

M3 - Chapter

AN - SCOPUS:85096281138

SN - 9789811202384

SP - 1561

EP - 1598

BT - Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning (In 4 Volumes)

PB - World Scientific Publishing Co.

ER -