In the Wikipedia entry for the Brownian bridge it says that
$$ B(t) = \frac{(T-t)}{\sqrt T} W\left(\frac{t}{T-t}\right) $$
where $B$ is a Brownian bridge starting at time $0$ at the origin and ending at time $T$ at the origin, and $W$ is the standard Wiener process (only constrained to start at time $0$ at the origin).
I was wondering what is the analogous conversion between the Wiener process and the pinned Brownian motion?
Say that $P$ is a pinned Brownian motion, constrained to end at time $T$ at position $L$. Then is it correct to write $$ P(t) = \frac{(T-t)}{\sqrt T} W\left(\frac{t}{T-t}\right) + \frac{t}{T} L \,? $$
I just want to make sure I am not missing anything.