Synonymity, generalized martingales, and subfiltrations. (English) Zbl 0545.60040

Hoover, Douglas N.

Ann. Probab. 12, 703-713 (1984).

The notion of synonymity of two processes X and Y refines the notion of ”X and Y having the same law” by taking into account the relation of the processes to their underlying filtrations. Synonymity was introduced by D. J. Aldous [Ecole d’Été de Saint-Flour 1983, to appear in Lect. Notes Math.] and further investigated by the author and H. J. Keisler [Adapted probability distributions. Trans. Am. Math. Soc. 286, 159-201 (1984)].
In this paper the author shows that generalized martingale properties, such as the semimartingale property, are preserved under synonymity, and that synonymous semimartingales have decompositions with the same distribution law. [For interesting applications of synonymity to stochastic differential equations see the author and E. Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations I,II, ibid. 275, 1-58 (1983)]. Furthermore the author gives a relatively elementary proof of the theorem (due to C. Stricker) that a semimartingale remains a semimartingale with respect to any subfiltration to which it is adapted.

Reviewer: M.Dozzi

MSC:

60G07	General theory of stochastic processes
60G48	Generalizations of martingales

Keywords:

synonymity; semimartingale; decompositions