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

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.

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