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The Setup:

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 $t \in [0, 1]$.

The Problem:

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.

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1 Answer

If this can be of any help, the probability that the bridge crosses $f(t)$ on $[0,1]$ is given by $$ \mbox{exp}(-2b(b+m)), $$ you can find out more and proof in this article by Scheike.

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