Here is an argument that the law of a Brownian motion conditioned to have a maximum at a certain level is a Bessel process of order 3. Let W be a one-dimensional Brownian motion killed on hitting 0....

Here is an argument that the law of a Brownian motion conditioned to have a maximum at a certain level is a Bessel process of order 3. Let W be a one-dimensional Brownian motion killed on hitting 0. Let
  be the maximum. By Exercise 19.1, X = (W, S) is a Markov process. Determine the law of X for
  where L is the last time X hits the diagonal. To define L more precisely, let
  s finite, a.s., because W will hit 0 in finite time with probability one.

Chapter 23