In this post I will be concerned with the following problem — given a martingale X for which we know the distribution at a fixed time, and we are given nothing else, what is the best bound we can obtain for the maximum of X up until that time? This is a question with a long history, starting with Doob’s inequalities which bound the maximum in the norms and in probability. Later, Blackwell and Dubins (3), Dubins and Gilat (5) and Azema and Yor (1,2) showed that the maximum is bounded above, in stochastic order, by the HardyLittlewood transform of the terminal distribution. Furthermore, this bound is the best possible in the sense that there do exists martingales for which it can be attained, for any permissible terminal distribution. Hobson (7,8) considered the case where the starting law is also known, and this was further generalized to the case with a specified distribution at an intermediate time by Brown, Hobson and Rogers (4). Finally, HenryLabordère, Obłój, Spoida and Touzi (6) considered the case where the distribution of the martingale is specified at an arbitrary set of times. In this post, I will look at the case where only the terminal distribution is specified. This leads to interesting constructions of martingales and, in particular, of continuous martingales with specified terminal distributions, with close connections to the Skorokhod embedding problem.
I will be concerned with the maximum process of a cadlag martingale X,
which is increasing and adapted. We can state and prove the bound on relatively easily, although showing that it is optimal is more difficult. As the result holds more generally for submartingales, I state it in this case, although I am more concerned with martingales here.
Theorem 1 If X is a cadlag submartingale then, for each and ,

(1) 
Proof: We just need to show that the inequality holds for each , and then it immediately follows for the infimum. Choosing , consider the stopping time
Then, and whenever . As is nonnegative and increasing in z, this means that is bounded above by . Taking expectations,
Since f is convex and increasing, is a submartingale so, using optional sampling,
Letting increase to gives the result. ⬜
The bound stated in Theorem 1 is also optimal, and can be achieved by a continuous martingale. In this post, all measures on are defined with respect to the Borel sigmaalgebra.
Theorem 2 If is a probability measure on with and then there exists a continuous martingale X (defined on some filtered probability space) such that has distribution and (1) is an equality for all .
(more…)