Exploding Levy processes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T22:20:23Z http://mathoverflow.net/feeds/question/14762 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14762/exploding-levy-processes Exploding Levy processes Grzenio 2010-02-09T11:45:48Z 2010-09-13T01:39:15Z <p>Hi, </p> <p>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.</p> http://mathoverflow.net/questions/14762/exploding-levy-processes/15575#15575 Answer by The Bridge for Exploding Levy processes The Bridge 2010-02-17T12:44:27Z 2010-02-17T12:44:27Z <p>Hi,</p> <p>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. </p> <p>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.</p> <p>(for references, google at "Black-Karasinski short rate model")</p> <p>Regards</p> http://mathoverflow.net/questions/14762/exploding-levy-processes/38523#38523 Answer by Steven Heston for Exploding Levy processes Steven Heston 2010-09-13T01:39:15Z 2010-09-13T01:39:15Z <p>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.</p>