# how to get brownian bridges maximum and minimum's joint distribution

I'm wondering how to get brownian bridge's maximum and minimum's joint distribution.Brownian bridge's definition is as below.

$$B_t = W_t - tW_1$$ for $$t \in [0,1]$$, W is wiener process

How to get it? In fact, I didn't get maximum's distribution of brownian bridge. I cannot catch any idea.. I tried to get this distribution via conditional probability, that fix $$W_1$$, but it is still hard one. Can you help me?

edit) I think that it is concerned with joint distribution of brownian motion's maximum and minimum, and W(1). Then how to attain this one?

Let $$M$$ and $$m$$ be the Max and min of $$B_t$$. Let us find the joint cdf of $$M$$ and $$-m$$, defined by $$P(\{M< a\}\cap \{-m< b\}),$$ In fact, the argument you can use is exactly the continuous version of the discrete problem here: A generalization of reflection. Namely, Brownian motion is the limit of a discrete random walk, and there is an exact formula for the number of random walks of a given length who stay between two horizontal lines, using the reflection principle.
Applying that logic, we get that $$P(M Note that this does not quite make sense, as all the probabilities are zero, but you have to understand this as a limit. Anyways, what you get is $$P(M This gives the joint cdf of $$M$$ and $$-m$$.