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ENDOGENOUS DOWNWARD JUMP DIFFUSION AND BLOW UP PHENOMENA BEFORE CRASH

  • Kwon, Young-Mee ;
  • Jeon, In-Tae ;
  • Kang, Hye-Jeong
  • Received : 2008.06.25
  • Published : 2010.11.30

Abstract

We consider jump processes which has only downward jumps with size a fixed fraction of the current process. The jumps of the pro cesses 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 shold explode in finite time, say $t_e$, conditional on no crash For the case of $\lambda$(X) = (lnX)$^{\alpha}$, 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

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