Is There a Predictable Criterion for Mutual Singularity of Two Probability Measures on a Filtered Space?

The theme of providing predictable criteria for absolute continuity and for mutual singularity of two density processes on a ltered probability space is extensively studied, e.g., in the monograph by J. Jacod and A. N. Shiryaev [JS]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical di culty remained open ([JS], p210): \We do not know whether it is possible to derive a predictable criterion (necessary and su cient condition) for P 0 T ? PT , : : : ". It turns out that to this question raised in [JS] which we also chose as the title of this note, there are two answers: 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 di culty comes from the fact that the density process of a probability measure P with respect to another measure P 0 may suddenly jump to zero. On the positive side we can characterize the set, where P 0 becomes singular with respect to P | provided this does not happen in a sudden but rather in a continuous way | as the set where the Hellinger process diverges, which certainly is a \predictable criterion". This theorem extends results in the book of J. Jacod and A. N. Shiryaev [JS].