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On spectral analysis of heavy-tailed Kolmogorov- Pearson diffusions

The self-adjointness of the semigroup generator of one dimensional diffusions implies a spectral representation (see [33, 50]) which has found many useful applications, for example for the prediction of second order stationary sequences (see [18]) and in mathematical finance (see [47]). However, on noncompact state spaces the spectrum of the generator will typically include both a discrete and a continuous part, with the latter starting at a spectral cutoff point related to the nonexistence of stationary moments. The significance of this fact for statistical estimation is not yet fully understood. We consider here the problem of spectral representation of transition density for an interesting class of examples: the hypergeometric diffusions with heavytailed Pearson type invariant distribution of a) reciprocal (inverse) gamma, b) Fisher – Snedecor, or c) skew-Student type. As opposed to the “classic” hypergeometric diffusions (Ornstein – Uhlebeck, Gamma/CIR, Beta/Jacobi), these diffusions have a continuum spectrum, whose spectral cutoff and transition density we present in this paper.

diffusion process; infinitesimal generator; Kolmogorov – Pearson diffusion; Sturm – Liouville equation; transition density

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