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Let $\{W'_t\}_{t\in [0,1]}$ be the Brownian motion on the real line obtained by taking the standard Brownian motion $\{W_t\}_{t\ge 0}$ and conditioning on the events $W_1 = 1$ and $0\le W_t\le 1$ for $t\in[0,1]$. My first question is whether $W'$ has a name.

Let $T$ be chosen uniformly at random in $[0,1]$. Secondly, is it possible to calculate explicitly the p.d.f. of $W'_T$?. In words, I am wondering how much does $W'$ favor the middle parts of $[0,1]$ and whether this can be calculated explicitly.

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    $\begingroup$ Assuming $W_t$ starts at $W_0=0$, the event $W_t\geq 0$ for $t\in[0,1]$ has probability zero, so $W'$ is not defined. $\endgroup$ – John Pardon Nov 11 '17 at 23:56
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    $\begingroup$ Check Brownian excursion $\endgroup$ – OOESCoupling Nov 12 '17 at 0:06
  • $\begingroup$ Thanks a lot. I think I have phrase this question as the limiting case of the discrete random walks of length tn by equipartitioning [0,1] into n pieces. $\endgroup$ – boinkboink Nov 12 '17 at 0:06
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    $\begingroup$ This conditioning can be made rigorous using a space-time Doob $h$-transform. Aside from the usual references, you can find some nice examples at toywiki.xyz/doob_transform.html $\endgroup$ – HMPanzo Nov 12 '17 at 0:42
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  • Without the condition $0\le W_t\le1$ on $[0,1]$, it calls a Brownian bridge: $$ W'_T \sim B_T + T, $$ where $B=\left(B_t\right)_{t\ge0}$ is the standard Brownian motion.
  • And with the condition, it has probability $0$ to occur: the Brownian motion will take negative values around $t=0$ with probability $1$ (consequence of Proposition 9, p. 28).

    Also, from the Arcsin Law:

    Theorem : The probability that the Brownian motion $\left(B\right)_{t\ge0}$ has no zeros in the time interval $(a,b)$ is given by $\frac{2}{\pi}\arcsin\sqrt{\frac{a}{b}}$.

    So for $a=0$, it is zero, and by independent increments the probability to remain positive in a neighborhood of $0$ is null.

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