In this note, the notion of an optimal transform of a (discrete parameter) stochastic process is introduced. Such transforms are shown to exist in certain cases, and a relationship to optimal stopping times is discussed. These ideas lead naturally to the representation of any given stochastic process as the transform of a submartingale. This type of representation theorem is extended to continuous parameter processes, where it is shown that in certain cases a quasi-martingale can be represented as a stochastic integral with respect to a submartingale.
"Transforms of Stochastic Processes." Ann. Math. Statist. 39 (2) 372 - 376, April, 1968. https://doi.org/10.1214/aoms/1177698398