Summary
We study the distance in variation between probability measures defined on a measurable space (Ω, ℱ) with right-continuous filtration (ℱt)t ≦0. To every pair of probability measures P and \(\tilde P\) an increasing predictable process \(h = h(P,{\text{ }}\tilde P)\) (called the Hellinger process) is associated. For the variation distance \(\left\| {P_T - \tilde P_T } \right\|\) between the restrictions of P and \(\tilde P\) to ℱ T (T is a stopping time), lower and upper bounds are obtained in terms of h. For example, in the case when \(P_0 = \tilde P_0 \),
In the cases where P and \(\tilde P\) are distributions of multivariate point processes, diffusion-type processes or semimartingales h are expressed explicitly in terms of given predictable characteristics.
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Kabanov, Y.M., Liptser, R.S. & Shiryaev, A.N. On the variation distance for probability measures defined on a filtered space. Probab. Th. Rel. Fields 71, 19–35 (1986). https://doi.org/10.1007/BF00366270
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DOI: https://doi.org/10.1007/BF00366270