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On the variation distance for probability measures defined on a filtered space
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  • Published: January 1986

On the variation distance for probability measures defined on a filtered space

  • Yu. M. Kabanov1,
  • R. Sh. Liptser2 &
  • A. N. Shiryaev3 

Probability Theory and Related Fields volume 71, pages 19–35 (1986)Cite this article

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  • 27 Citations

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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 \(P_0 = \tilde P_0 \),

$$2(1 - (E{\text{ }}\exp {\text{ }}( - h_T ))^{1/2} ) \leqq \left\| {P_T - \tilde P_T } \right\| \leqq 4(Eh_T )^{1/2} $$

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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Authors and Affiliations

  1. Central Economical-Mathematical Institute, Krasikova 32, Moscow, USSR

    Yu. M. Kabanov

  2. Institute of Control Science, Profsojuznaja 65, Moscow, USSR

    R. Sh. Liptser

  3. Mathematical Institute of Academy of Science, Vavilova 42, Moscow, USSR

    A. N. Shiryaev

Authors
  1. Yu. M. Kabanov
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  2. R. Sh. Liptser
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  3. A. N. Shiryaev
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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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  • Received: 20 April 1984

  • Accepted: 26 April 1985

  • Issue Date: January 1986

  • DOI: https://doi.org/10.1007/BF00366270

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Keywords

  • Filtration
  • Stochastic Process
  • Probability Measure
  • Probability Theory
  • Mathematical Biology
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