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Theory of Probability and its Applications / Volume 44 / Issue 1

Theory Probab. Appl. 44, pp. 51-59 (9 pages)

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

W. Schachermayer and W. Schachinger

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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.

© 2000 Society for Industrial and Applied Mathematics

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0040-585X (print)  
1095-7219 (online)

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$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

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http://dx.doi.org/10.1137/S0040585X97977367

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