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The problem I am trying to solve is the following:

What is the expected time the Wiener process $W_{t}$ stays above t-axis for $t\in [0;1]$ if we know that $W_{1}=a$?

I suppose that this is a well known problem, but I can't find a useful reference. If there is no closed form solution for this problem but maybe there are some approximations?

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2 Answers 2

up vote 3 down vote accepted

The pinned Brownian motion $W^a_t$ can be expressed in terms of usual Brownian motion $W_t$ by $W^a_t=at+(1-t) W_{t/(1-t)}$. Thus, for any fixed $0\leq t\leq 1$, $$\mathbb{P}(W^a_t\geq 0)=\mathbb{P}\left( W_{t/(1-t)} \geq {-at\over 1-t}\right)={1\over 2}+{1\over 2}\mbox{erf}\left({a\over\sqrt{2}} \sqrt{t\over 1-t}\right).$$

The expected time that $W^a_t$ is positive will be the integral of this function over $0\leq t\leq 1$. I don't think it can be simplified.

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Thanks a lot, Byron! –  Roah Apr 26 '11 at 15:47
@Roah No problem. I hope it helps. –  Byron Schmuland Apr 26 '11 at 15:52
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The integral suggested by Byron can be pushed around a little, leading to the expression $$ 1-{1\over 2} E[Z^2/(a^2+Z^2)]={1\over 2}+{1\over 2}E[a^2/(a^2+Z^2)] $$ for the mean time spent above the $t$-axis by the pinned Wiener process. Here $Z$ is a standard normal random variable.

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