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Let's say I have a Brownian motion, such that I know its value at time 0 (0) and time T (also 0). I am trying to evaluate the time spent above 0 between time 0 and T.

Obviously I know that the average of this value is 1/2, but is there a way to know the full law of that duration? Or are there other known theoretical results for this? I am guessing this is a fairly common problem but I am unable to find references for this.

Thanks!

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2 Answers 2

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Yes the time spent above zero by the trajectory of a Brownian bridge is a uniform random variable, see Exercise 3.9 in Chapter 12 of Continuous Martingales and Brownian Motion by Revuz and Yor.

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Did told me this is a well known result, though Googling confirms that it is not very easy to find on the internet (as you probably realised)

For a Brownian Bridge $X$ with $X_0=0=X_1$, the distribution is a Uniform distribution on $[0,1]$. This result is due to Levy. I found this in a book called Handbook of Beta distribution and its applications page 193.

(I was able to view the part of this book for this answer on Google.)

Notice the Arcsin law for a Brownian motion says the amount of time spent by a Brownian motion above $0$ is Beta distribution with parameters $(1/2, 1/2)$ and uniform distribution is Beta distribution with parameter $(1,1)$.

The book claims the results has since been generalised, but does not give any references.

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