A decomposition theorem for supermartingales

Abstract

It is a well known fact (see [1], p. 296) that any discrete-parameter super- martingale Xn} ndg can be represented as a sum:X,= Y,+Z,, Y} being martingale, and {Zn} a process with decreasing sample functions, such that Z0 0.Moreover, if the supermartingle {Xn} is uniformly in- tegmble, the same is true for Yn} and {Zn}.Doob has raised the problem of the existence of such a decomposition for continuous-parameter supermartin- gales.We shall solve this problem here, lthough the necessary and suffi- cient condition we give is not very easy to handle.Our proof has been adapted from that of a theorem in potential theory, concerning the representation of excessive functions as potentials of dditive functionls ([3], pp.75-83).A reader with some knowledge of Hunt's potential theory for Markov proc- esses, nd the theory of dditive functionals, will easily recognize here some kind of a coarse potential theory, with supermrtingales replacing excessive functions.Our terminology hs been chosen in accordance with this ide.We shall use freely the results contained in Chapter VII (mrtingale theory) of Doob's book.A number of definitions will be recalled, for the reader's convenience.1. Let 2 be a set, a Borel field of subsets of , P a probability measure defined on (, if).We are given a family {} ,R+ of Borel subfields of if, such that , s t t any process which is well adapted to the t family is well adapted to the t+ family.A supermartingale (relative to the t family) is a real valued process {Xt}, well adapted to the t family, such that (ii) V,, ,, [X,+t I,] =< X a.s.If equality holds a.s. in (ii), the process is a martingale.We shall be concerned here only with sample right continuous supermartingales.If {X} is such a supermartingale, then (ii) holds with :t replaced by +.Let indeed A be an event in ff,+, and let s be a decreasing sequence which converges to s.

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