A characterization of adapted distribution. (English) Zbl 0634.60033

Hoover, Douglas N.

Ann. Probab. 15, 1600-1611 (1987).

Author’s abstract: The notion of adapted distribution of a stochastic process was introduced in a recent paper of the author and H. J. Keisler [Trans. Am. Math. Soc. 286, 159-201 (1984; Zbl 0548.60019)]. Here we give a simple characterization of this notion in terms of filtration embeddability. This characterization allows us to show that for a local martingale M for which some ordinary stochastic differential equation \(X_ t=\int^{t}_{0}f(s,X_ s)dM_ s\) admits sufficient nonuniqueness in law of the solutions X, the class of possible joint laws of (M,X) determines the adapted law of M.

Reviewer: J.A.Goldstein

MSC:

60G07	General theory of stochastic processes
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

adapted distribution of a stochastic process; filtration embeddability; local martingale; nonuniqueness in law

Citations:

Zbl 0548.60019