Every Nonnegative Submartingale is the Absolute Value of a Martingale

Abstract

It is shown that every nonnegative superfair process (in particular a nonnegative submartingale) is the absolute value of a symmetric fair process (martingale). Is every submartingale a convex function of a martingale? No. If however the adjective convex is omitted from the question, an affirmative answer is provided. Furthermore, transforming functions $\phi$, such that every superfair process (submartingale) is that $\phi$ of a fair process (martingale), are shown to exist. The results are extended to continuous-parameter submartingales with rightcontinuous sample functions.