  Monotone processes, just like martingales, can often be recovered from their
final values. Examples include running maxima of supermartingales, as well as
running maxima, local times, and various integral functionals of sticky
processes such as fractional Brownian motion. An interesting corollary is that
any positive local martingale can be reconstructed from its final value and its
global maximum. These results rely on the notion of conditional infimum, which
is developed for a large class of complete lattices. The framework is
sufficiently general to handle also more exotic examples, such as the process
of convex hulls of certain multidimensional processes, and the process of sites
visited by a random walk.
