Doob's decomposition theorem for near-submartingales

We study the discrete parameter case of near-martingales, nearsubmartingales, and near-supermartingales. In particular, we prove Doob’s decomposition theorem for near-submartingales. This generalizes the classical case for submartingales. 1. Motivation From Non-adapted Stochastic Integral Let B(t), t ≥ 0, be a Brownian motion starting at 0 and {Ft} the filtration given by B(t), namely, Ft = σ{B(s); 0 ≤ s ≤ t}, t ≥ 0. The Itô integral ∫ b a f(t) dB(t) (see, e.g., the book [8]) is defined for {Ft}-adapted stochastic processes f(t) with almost all sample paths being in L[a, b]. Several extensions of the Itô theory of stochastic integration to cover non-adapted integrands have been introduced and extensively studied by, just to mention a few names, Buckdahn [3], Dorogovtsev [4], Hitsuda [5], Itô [6], Kuo–Potthoff [10], León–Protter [12], Nualart–Pardoux [13], Pardoux–Protter [14], Russo–Vallois [15], and Skorokhod [16]. In particular, in his lecture for the 1976 Kyoto Symposium, Itô [6] gave rather elegant ideas to define the following non-adaptive stochastic integral