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probably this is a fairly newbie question, but is it possible that the a generic Levy process explodes (i.e. tends to infinity for finite time t with positive probability)? If yes, could you please provide an example, if no point me to the proof.

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As remarked by Leonid Kovalev a Lévy process doesn't explodes as far as I know, nevertheless as you mention generic Levy Process for which I don't know any definition, you may be thinking of them as diffusions driven by a Lévy process.

Then looking at some SDEs, you can have some cases where explosion time is a.s. finite, and even when the driving procsess is a Brownian Motion, for example I think I can remember that $X_t$ verifying $d[Ln(X_t)]=a(b-Ln(X_t))dt+\sigma.dW_t$, $X_0=0$ is of this type.

(for references, google at "Black-Karasinski short rate model")


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I think that by "generic Levy process" you simply mean a stochastic process with Levy noise instead of a diffusion. Consider the reciprocal of a process that can access the origin, such as a Wiener process or compensated Poisson process. Clearly when the original process hits zero, the reciprocal explodes.

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