Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not asking for the (conditional) probability distribution of $Y(t)$
I am thinking of this as a generalization of the Brownian bridge, which has several representation in terms of the Brownian motion $B(t)$.