**Walter Schachermayer**

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

**W. Schachinger, W. Schachermayer**

Theory of Probability and its Applications, Vol. 44 (1999), No. 1, pp. 51-59.

### Abstract:

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.

#### Keywords:

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

### Preprints:

[PostScript (123 k)] [PS.gz (47 k)] [PDF (191 k)] [DOI: 10.1137/S0040585X97977367]

Publications marked with have appeared in refereed journals.

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