Doob-Meyer decomposition theorem ($X=M+A$)

Under the usual filtration conditions, a càdlàg submartingale of Class D on a finite horizon has a unique decomposition into a uniformly integrable martingale and a predictable integrable nondecreasing process starting at zero. A general continuous semimartingale decomposition instead allows an arbitrary continuous finite-variation drift. A convex function of a semimartingale need not be a submartingale when that drift has unrestricted sign.