For the study of the paths of stochastic processes, the upcrossing-down-crossing arguments are available in the real-valued case but not in the vector-valued case. We use in this article stopping time methods and the Kadec-Klee renorming theorem to obtain regularity properties for Banach-lattice valued submartingales and Banach-valued pramarts.
Nikos E. Frangos. "On Regularity of Banach-Valued Processes." Ann. Probab. 13 (3) 985 - 990, August, 1985. https://doi.org/10.1214/aop/1176992919