Cumulative probability that random walk variable exceeds given value

*To*: mathgroup at smc.vnet.net*Subject*: [mg104834] Cumulative probability that random walk variable exceeds given value*From*: Kelly Jones <kelly.terry.jones at gmail.com>*Date*: Wed, 11 Nov 2009 04:28:02 -0500 (EST)

How can I use Mathematica to solve this problem? Let x[t] be a normally-distributed random variable with mean 0 and standard deviation Sqrt[t]. In other words, x[0] is 0, x[1] follows the standard normal distribution, x[2] follows the normal distribution with mean 0 and standard deviation Sqrt[2], etc. It's easy to compute the probability that x[5] > 2 (for example). How do I compute the probability that x[t] > 2 for 0 <= t <= 5. In other words, the probablity that x[t] surpassed 2 at some point between t=0 and t=5, even though x[5] may be less than 2 itself. Notes: % My goal: predicting whether a continuous random walk will exceed a given value in a given period of time. % I realize that saying things like "x[5] may be less than 2" is sloppy, since x[5] is a distribution, not a value. Hopefully, my meaning is clear. % I tried doing this by adding/integrating probabilities like this (psuedo-code): P(x[t] > 2 for 0 <= t <= 5) = Integral[P(x[t] > 2),{t,0,5}] but this overcounts if x[t] > 2 for multiple values of t. -- We're just a Bunch Of Regular Guys, a collective group that's trying to understand and assimilate technology. We feel that resistance to new ideas and technology is unwise and ultimately futile.

**Follow-Ups**:**Re: Cumulative probability that random walk variable***From:*Leonid Shifrin <lshifr@gmail.com>

**Re: Cumulative probability that random walk variable***From:*DrMajorBob <btreat1@austin.rr.com>

**Re: Cumulative probability that random walk variable exceeds***From:*Daniel Lichtblau <danl@wolfram.com>