Is there a predictable criterion for mutual singularity of two probability measures on a~filtered space?

W. Schachermayer ; W. Schachinger

Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 101-110

Résumé

The theme of providing predictable criteria for absolute continuity and for mutual singularity of two density processes on a filtered probability space is extensively studied, e.g., in the monograph by J. Jacod and A. N. Shiryaev [Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open [J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987, p. 210]: "We do not know whether it is possible to derive a predictable criterion (necessary and sufficient condition) for $P_T'\perp P_T,\ldots$". It turns out that there are two answers to this question raised in the monograph of J. Jacod and A. N. Shiryaev: On the negative side, we give an easy example showing that in general the answer is no, even when we use a rather wide interpretation of the concept of “predictable criterion”. The difficulty comes from the fact that the density process of a probability measure $P$ with respect to another measure $P'$ may suddenly jump to zero. On the positive side, we can characterize the set where $P'$ becomes singular with respect to $P$—provided this happens in a continuous way rather than suddenly—as the set where the Hellinger process diverges, which certainly is a "predictable criterion." This theorem extends results in the monograph of J. Jacod and A. N. Shiryaev.

Détail

Keywords: continuity and singularity of probability measures, Hellinger processes, stochastic integrals, stopping times.