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Polar sets of anisotropic Gaussian random fields

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  • Jakob Söhl

Abstract

This paper studies polar sets of anisotropic Gaussian random fields, i.e. sets which a Gaussian random field does not hit almost surely. The main assumptions are that the eigenvalues of the covariance matrix are bounded from below and that the canonical metric associated with the Gaussian random field is dominated by an anisotropic metric. We deduce an upper bound for the hitting probabilities and conclude that sets with small Hausdorff dimension are polar. Moreover, the results allow for a translation of the Gaussian random field by a random field, that is independent of the Gaussian random field and whose sample functions are of bounded Hölder norm.

Suggested Citation

  • Jakob Söhl, 2009. "Polar sets of anisotropic Gaussian random fields," SFB 649 Discussion Papers SFB649DP2009-058, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2009-058
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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2009-058.pdf
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    Keywords

    Anisotropic Gaussian fields; Hitting probabilities; Polar sets; Hausdorff dimension; European option; Jump diffusion; Calibration;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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