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I know from Karatzas & Shreve (1991) that a Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[0,T]$ satisfies: $B(t)=a(1-t/T) + b*t/T + [W(t) - W(T)*t/T]$, where $W(t)$ is a standard one-dimensional Brownian motion.

By the above equation we can get its distribution. My question is what's the distribution of the Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[T_1, T_2]$?

Any idea or reference?

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The starting point doesn't matter. You're conditioning on the Brownian Motion being $a$ at time $T_1$, so there's no variance there. Just do the calculation on $[0, T_2 - T_1]$. –  quasi Apr 22 at 23:14
    
Thanks for your reply. But what about t? t should be on time interval [T1, T2]. So I think do the calculation with t -> t-T1 and T -> T2-T1. –  Jack Apr 30 at 21:13

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