Let X be a discrete time process on some (ohm, F, P), and let F = {Fn } n>=0 be a filtration. Assume that X is adapted, X0 = 0 and that Xn is integrable for all n. Define the process M by M0 = 0 and Mn = Mn-i + Xn - E [Xn|Fn-1 for n >= 1. Show that M is a martingale. Define then A = X - M. Show that An is Fn-1 -measurable for all n >= 1 (one says that A is predictable) and that A0 = 0. Show that A is increasing iff X is a submartingale