Suppose I want to generate a random Brownian motion $B$ on $[0,1]$ such that:
- $B_0=x_0$
- $B_1=x_1$
- $\max B_t = M$
- $\min B_t = m$
The first two conditions a not difficult to impose. However I have trouble incorporating the 3th and 4th conditions, especially the fact that $M$ and $m$ have to actually be attained (rather then $M,m$ only being bounds).
Of course I want to do this in such a way that the probability of generating a specific walk is the correct conditional probability (as induced by the standard pdf on the set of Brownian motions).
