A Brownian Bridge $B$ is a Brownian Motion on time interval $[0, 1]$ conditioned such that $B(0) = B(1) = 0$. I have a function $f(t) = mt+b$ with $m, b$ set such that $C(t) \le 0$ for $t \in [0, 1]$. Define a new process $C$ that is a Brownian Bridge conditioned such that $C(t) \ge f(t)$ for any $t \in [0, 1]$.
I am interested in finding a density function for the random variable $\int_0^1 C(t) dt$.
A Possibly Useful Fact:
If $f(t) = 0$ identically, then $C$ is called a Brownian Excursion Process, and the density function for its integral is known.