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Does anyone know how to compute $\text{Cov}[\max_{s\in [0,1]}B(s), B(t)]$ where $B(t)$ is the standard Brownian bridge on the interval $[0,1]$?

Update. I have found a paper that solves the problem: On the maximum of the generalized brownian bridge (Beghin, Orsingher)

Mark G.
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    I haven't worked this out, but it should be possible to do this with Girsanov. Use the fact that $B(t)$ solves $dB = - \frac{B}{1-t}dt + dW$ on $0 \leq t < 1$ and use Girsanov to rewrite this as a BM. Since the joint distribution of BM and the maximum of BM is known, this reduces the problem to a (possibly computable) integral. – Chris Janjigian Apr 23 '14 at 00:13
  • Thank you @ChrisJanjigian. Unfortunately I am a novice in the sector and a lot of literature is already present. Do you suggest any particular reference to look at? – Mark G. Apr 24 '14 at 16:51

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