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Monge–Kantorovich Mass Transference Problem, Minimal Distances and Minimal Norms

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Abstract

The goals of this chapter are to: Introduce the Kantorovich and Kantorovich–Rubinstein problems in one-dimensional and multidimensional settings; Provide examples illustrating applications of the abstract problems; Provide examples illustrating applications of the abstract problems; Discuss the multivariate Kantorovich and Kantorovich–Rubinstein theorems, which provide dual representations of certain types of minimal distances and norms; Discuss a particular application leading to an explicit representation for a class of minimal norms.

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Notes

  1. 1.

    The program of the conference and related materials are available online at http://www.mccme.ru/%5C~ansobol/otarie/MK2012conf.html.

  2. 2.

    See Rachev [1991], Rachev and Taksar [1992], Rachev and Hanin [1995a,b], Cuesta et al. [1996], and Rachev and Rüschendorf [1999].

  3. 3.

    See, for example, Bazaraa and Jarvis [2005].

  4. 4.

    See also the general discussion in Whittle (1982, p. 210–211).

  5. 5.

    See Gray (1988, p. 48).

  6. 6.

    See Kalashnikov and Rachev [1988, Chaps. 3 and 6].

  7. 7.

    See, for example, Bazaraa and Jarvis [2005].

  8. 8.

    See Bazaraa and Jarvis [2005] and Berge and Chouila-Houri [1965, Sect. 9.8].

  9. 9.

    See Fortet and Mourier [1953].

  10. 10.

    See Dunford and Schwartz [1988, p. 65].

  11. 11.

    See also Dudley [2002, Theorem 11.6.2].

  12. 12.

    See Sect. 2.2 of Chap. 2.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics College of Business, Stony Brook University, Stony Brook, NY, USA

    Svetlozar T. Rachev

  2. FinAnalytica, New York, NY, USA

    Svetlozar T. Rachev

  3. Department of Probability and Statistics, Charles University, Prague, Czech Republic

    Lev B. Klebanov

  4. EDHEC Business School EDHEC-Risk Institute, Singapore, Singapore

    Stoyan V. Stoyanov

  5. EDHEC Business School EDHEC-Risk Institute, Nice, France

    Frank J. Fabozzi

Authors
  1. Svetlozar T. Rachev
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  2. Lev B. Klebanov
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  3. Stoyan V. Stoyanov
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  4. Frank J. Fabozzi
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Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., Fabozzi, F.J. (2013). Monge–Kantorovich Mass Transference Problem, Minimal Distances and Minimal Norms. In: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4869-3_5

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