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Can anyone point me to the expression for the first passage time for a geometric Brownian motion process X(t) as a function of the starting point, threshold, drift and diffusion parameters.

I am mainly interested for processes with positive drift and thresholds that are higher than the starting point.

I know this is a standard expression, however all results I have found so far are specific to a particular starting point (e.g. X(0) = 0) and/or threshold of 1, and I am not sure how to generalize to any initial state and threshold value.

Any help would be much appreciated

Thanks

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    $\begingroup$ Passage time distributions for (arithmetic) Brownian motion are available in a lot of places, i.e. Karatzas/Shreve. Just take logs to reduce to that problem. $\endgroup$
    – quasi
    Aug 28, 2013 at 23:23
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    $\begingroup$ Are you sure the solutions found deal with a GBM starting point at 0? :-) $\endgroup$
    – Quartz
    Aug 29, 2013 at 8:46
  • $\begingroup$ Could you please share the explicit expression for the pdf of the FPT of the GBM, if you found it? Thank you $\endgroup$
    – Confounded
    Sep 20, 2021 at 14:48

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Just work in log-space to get rid of the starting point, then by invariance of BM you only need threshold-X(0) and X(0)=0 is enough to work with at first. In the no-drift case the solution is also invariant if you scale diffusion and threshold simultaneously (Levy dist), therefore you can effectively get rid of one parameter (that is you can reduce to the case of it being =1). In the general case (IG) you can imagine a similar behaviour acting on the drift too (so a combined drift and threshold scaling must be "compensated" by one in volatility), and again you get rid of one parameter, in this case the threshold.

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In log space the explicit solution for the density of the first passage time is the Inverse Gaussian Distribution. See, e.g., http://www.springerreference.com/docs/html/chapterdbid/205395.html or the Wikipedia page for the distribution. The only thing that should matter is the interval from the initial state to the threshold, and that is the parameter "a" in the form given in the link above.

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