W. Schachinger, W. Schachermayer
Theory of Probability and its Applications, Vol. 44 (1999), No. 1, pp. 51-59.
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 [JS]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open ([JS], p210): "We do not know whether it is possible to derive a predictable criterion (necessary and sufficient condition) for $P_T'\perp P_T$,...". 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 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 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].
[JS] --- J. Jacod, A. N. Shiryaev: Limit Theorems for Stochastic Processes. Berlin: Springer 1987.
continuity and singularity of probability measures, Hellinger processes, stochastic integrals, stopping times
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