Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2010; 47(6): 1105-1119

Printed November 1, 2010

https://doi.org/10.4134/BKMS.2010.47.6.1105

Copyright © The Korean Mathematical Society.

Endogenous downward jump diffusion and blow up phenomena before crash

Youngmee Kwon, Intae Jeon, and Hye-Jeong Kang

Hansung University, Catholic University of Korea, Seoul National University

Abstract

We consider jump processes which has only downward jumps with size a fixed fraction of the current process. The jumps of the processes are interpreted as crashes and we assume that the jump intensity is a nondecreasing function of the current process say $\lambda(X)$ ($X=X(t)$: process). For the case of $\lambda(X)=X^{\alpha}, \alpha>0$, we show that the process $X$ should explode in finite time, say $t_e$, conditional on no crash. For the case of $\lambda(X)=(\ln X)^{\alpha}$, we show that $\alpha =1$ is the borderline of two different classes of processes. We generalize the model by adding a Brownian noise and examine the blow up properties of the sample paths.

Keywords: crash, explosion, jump diffusion

MSC numbers: 60J60, 60J45

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