Consider a particle undergoing geometric brownian motion with drift $\mu$ and volatility $\sigma$ e.g. as in here. Let $W_t$ denote this geometric brownian motion with drift at time $t$. I am looking for a formula to calculate:
$$ \mathbb{P}\big(\max_{0 \leq t \leq n} W_t - \min_{0\leq t \leq n} W_t > z\big) $$ The inputs to the formula will be $\mu$, $\sigma$, $z$, and $n$.
Given particle undergoing Geometric Brownian Motion, want to find formula for probability that max-min > z after n days
12
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probability
stochastic-processes
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2[Here](http://alumni.caltech.edu/~amir/drawdown-jrnl.pdf) is a result for (nongeometric) Brownian motion with drift for a related quantity. – 2011-03-12
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0I am going through that paper and it seems to be plain wrong. E.g.: try calculating G(0). It should give 1 since Prob(max-min>=0) = 1, but it leads to problems. Also, in the paper: E[D] = \int dhG(h). This is plain wrong. E[D] = \int hg(h)dh. – 2011-03-12
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0That's entirely possible. I haven't read that paper carefully. When I saw your question, I remembered having seen it maybe five or six years ago, so I typed it in to Google and linked to it. Caveat lector, I suppose. (Note, you can use standard TeX notation even in comments by wrapping math with `$`. – 2011-03-13
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0Also, let $B_t = \log W_t$ be a brownian motion with drift. Then if you know $\mathbb{P}(\min_{0 \leq t \leq n} B_t > a, \max_{0 \leq t \leq n} B_t < b)$, then you could conceivably back out what you desire by transformation of variables. Now, the quantity I have written is equivalent to the distribution of the first exit time of brownian motion with drift from the strip $(a,b)$ for $a < b$. I imagine some fairly strong statement about this must be known, since the distribution of $(B_t, \max_{u \leq t} B_u)$ is known, has closed form, and is closely related to the first hitting time of $b$. – 2011-03-13
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0Thanks. I'll have a look. I came across it myself last night as well as I was looking around for an answer. But, I haven't gotten a chance to read it yet. – 2011-03-14
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3I am adding back my comment (sorry I deleted it). For RV subjected to brownian motion, equation 4.10 in this [paper](www-sop.inria.fr/members/Etienne.Tanre/publication/jtp.pdf) gives the answer – 2011-03-14
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0also see double lookbacks http://som.yale.edu/~hh78/lb_9612.pdf to derive using transformation of variables – 2011-10-09