An Almost Sure Approximation for the Predictable Process in the Doob–Meyer Decomposition Theorem

The Doob–Meyer decomposition theorem opened the way towards the theory of stochastic integration with respect to square integrable martingales and—consequently—semimartingales, as described in the seminal paper [7]. According to Kallenberg [4], this theorem is “the cornerstone of the modern probability theory”. It is therefore not surprising that many proofs of it are known. To the author’s knowledge, all the proofs heavily depend on a result due to Doléans-Dade [3], which identifies predictable increasing processes with “natural” increasing processes, as defined by Meyer [6]. In the present paper we develop ideas of another classical paper by K. Murali Rao [8] and construct a sequence of decompositions for which the superior limit is pointwise (in (t, ω)) equal to the desired one, and thus we obtain predictability in the easiest possible way. Let (Ω,F , {Ft}t∈[0,T ], P ) be a stochastic basis, satisfying the “usual” conditions, i.e. the filtration {Ft} is right-continuous and F0 contains all P -null sets of FT . Let (D) denote the class of measurable processes {Xt}t∈[0,T ] such that the family {Xτ} is uniformly integrable, where τ runs over all stopping times with respect to {Ft}t∈[0,T ]. One of the variants of the Doob–Meyer theorem can be formulated as follows.