I am interested in the solution of the following problem: for a fixed $n$, find
$$\sup\ \ \mathbb{E}\max_{1\le i<j\le n}|M_i-M_j|^p,$$ where supremum is taken over all the martingales $$M_1, \ M_2, \ \dots, \ M_n,$$ such that $ \ 0 \le M_n\le 1 \ $ a.e. I am mostly concerned with the case of $ \ p \ $ equal to $ \ 1 \ $ or $ \ 2$.
This problem looks as if it must have been solved a long time ago. Have You ever encountered it? I know that there are similar articles for continues time case (with cadlag assumptions) but I would like to find a discrete version as stated above.
I would be very grateful if someone would point me towards an appropriate book or paper.