Natural processes and Doob-Meyer decompositions over a probability gage space

AbstractThe notion of naturalness for L1-processes over a probability gage space is defined and the uniqueness of Doob decompositions of supermartingales is discussed. In particular, it is shown that if (Xt) is an Itô-Clifford stochastic integral martingale, then (Xt∗Xt) has a unique decomposition into the sum of an L1-martingale and an increasing L1-process null at t = 0