Building on Yemon, who suggests that the solution is some distribution of a hitting time, if we assume the threshold for the 'hit' is sqrt(T), and that variance, u, of the random variable is directly proportional to the square root of time, then the mean time for that variable to exceed sqrt(T) may equal k * u * sqrt(T) * Phi(1)/2, or approximately k * u * sqrt(T)/16sqrt(T) * 0.16, distributed lognormally.

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Building on Yemon, who suggests that the solution is some distribution of a hitting time, if we assume the threshold for the 'hit' is sqrt(T), and that variance, u, of the random variable is directly proportional to the square root of time, then the mean time for that variable to exceed sqrt(T) may equal k * u * sqrt(T) * Phi(1)/2, or approximately k * u * sqrt(T)/16, distributed lognormally.