Generalized Density Process of Distributions of Semimartingales with Independent Increments: Computation and Applications

Scientific Advisor - Prof. Alexander A. Gushchin
The first part of the report will be devoted to two representations of the generalized density process of the distributions of semimartingales with independent increments. This result generalizes a well-known formula for the density process in the case where one distribution is locally absolutely continuous with respect to another one, as well as a formula for the density process of the distributions of Lйvy processes without assumption of the local absolute continuity obtained by K. Sato. In the second part of the report some results based on the obtained representations will be demonstrated. These results are of indepedent interest. 1. It appears that time-averaging of the triplet transforms a semimartingale with independent increments into a Lйvy process which is “closer” to every Lйvy process than the original process. The “closer” the processes are, the closer their f-divergences are. This fact also has an equivalent formulation in terms of the comparison of the respective binary statistical experiments. 2. An equivalence criterion for binary statistical experiments formed by the distribution of a semimartingale with independent increments and the distribution of a Lйvy process will be announced.