Geometric Brownian Motion First-Passage Times in Log Space
Summary
The document explains how to generalize first-passage results for geometric Brownian motion when the starting value and threshold are not normalized to convenient constants. Taking logarithms turns the process into Brownian motion with drift, so the relevant distance is the gap between the initial log state and the log threshold. The response identifies the first-passage-time density in the drifted case as an inverse Gaussian distribution, while the zero-drift case reduces to a Lévy distribution.
It also notes that scaling the threshold, drift, and diffusion together can reduce the number of independent parameters in the distribution. The discussion is conceptual and points to standard distribution results rather than deriving their density or cumulative probability formula. Its main stated interest is positive drift and an upper threshold, while applications to other directions or parameterizations require care about the sign convention and whether the barrier is reachable.
Key ideas
- Logging a geometric Brownian motion converts it into Brownian motion with drift.
- The first-passage distribution depends on the log-distance from the starting state to the threshold.
- For nonzero drift, the document identifies the first-passage density as inverse Gaussian.
- In the no-drift case, the corresponding hitting-time law is Lévy, with scaling invariance reducing parameters.
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# Brownian motion - first passage time # Brownian motion - first passage time 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 ## Answer by Quartz (score 2) https://quant.stackexchange.com/a/8851 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. ## Answer by user915 (score 2) https://quant.stackexchange.com/a/9053 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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