I would like to follow up on a previous question of mine that was asked in this forum earlier (Is $(\sup_{s\le t} B_s, B_t) $ a Markov process?)
I am trying to play with Markov processes and become more familiar with certain technicalities involved in dealing with them. Right now I am wondering about the best approach to find the transition probabilities for the Markov process $(B_t, M_t:=\sup_{0\le s \le t} B_s) $ where $B=\{B_t: t \in [0, \infty)\} $ is BM on a given filtered probability space.
For univariate processes, there might be a certain amount of algebra involved, but ideally I know how to attack the problem. Here, I do know the pdf for the joint probability of $(B_t, M_t) $ which is a standard result in the theory of BM and not even particularly complicated, but I feel a bit clumsy on how to organize my work to compute the transition probabilities. Again, any help would be greatly appreciated.
Thank you Maurice