Finitely Additive Supermartingales Are Differences of Martingales

It is shown that any nonnegative bounded supermartingale admits a Doob-Meyer decomposition as a difference of a martingale and an adapted increasing process upon appropriate choice of a reference probability measure which may be only finitely additive. Introduction. In [Armstrong, 1983] it is shown that every bounded finitely additive supermartingale is a decreasing process with respect to some reference probability measure P. This concept of a decreasing (or increasing) process is weaker than that corresponding to decreasing (increasing) processes of random variables adapted to a filtration. The corresponding class of finitely additive processes are called adapted decreasing (increasing) processes with respect to P. Theorem B asserts that for every bounded nonnegative finitely additive supermartingale Y there is a probability P so that Y — M A where M is a martingale and A is an adapted increasing process with respect to P. In order to establish this it is necessary in Proposition A to show that for g= {gt: te.T) an ordinary L^bounded nonnegative supermartingale adapted to the linearly ordered filtration ( ¿Ft: t e T} on the probability measure space (X, ¿F, P) to be expressed as the difference m — a where m is a martingale and a is an increasing process with 0 = inf, a, it is necessary and sufficient that {gT: t simple T-valued stopping time < /} be uniformly integrable for all t e T. This extends the usual Doob-Meyer Decomposition Theorem in allowing arbitrary linearly ordered T. Finitely additive supermartingales are differences of martingales and increasing adapted sequences. The Doob-Meyer Decomposition Theorem asserts that a nonnegative L1 -bounded supermartingale /={/,: 0<í<oo} adapted to a filtration (J^: 0 < t < oo} of sub-a-algebras in a probability space (X, ^, P) admits a decomposition / = m — a where m is a martingale and a is an increasing process with a0 = 0 iff / is of class DL. We recall that / is of class DL iff (/T: t stopping time < t} is uniformly integrable (i.e., a(Ll, L°°)-relatively compact) for each t e [0, oo ). Attention may be confined to stopping times t with only finitely many Received by the editors May 23, 1983 and, in revised form, April 16, 1984 and February 12, 1985. 1980 Mathematics Subject Classification. Primary 60G48, 28A60.