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

Yu Ting Chen, Cheng Few Lee, Yuan Chung Sheu

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

## 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 Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning (In 4 Volumes) World Scientific Publishing Co. 1561-1598 38 9789811202391 9789811202384 https://doi.org/10.1142/9789811202391_0041 Published - 1 Jan 2020

## Keywords

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