The joint law of a max-continuous local submartingale and its maximum

We consider the family of converging max-continuous local submartingales starting from zero. An equivalence relation for processes of this family is introduced so that two processes are called equivalent if their joint laws of the terminal value and the maximum are the same. We single out a subfamily of processes of simple structure that have a unique (in the sense of the distribution) representative in each equivalence class. Next, using an extension of Monroe's theorem, we embed a process of this subfamily in a Brownian motion using a minimal time-change, and from this embedding construct a continuous local martingale from the same equivalence class. Moreover, it is found that whether a process from this family belongs to the class of uniformly integrable martingales depends only on its equivalence class. So, these results provide an alternative approach to the problems of characterization of the distribution of a continuous local martingale and its maximum, which were considered by L. C. G. Rogers and P. Vallois in the early 1990s.