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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 which has found many useful applications, for example in mathematical finance. However, on non-compact 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 signifi- cance of this fact for statistical estimation is still not fully understood. We consider here the problem of spectral representation of the transition density for an interesting class of diffusions: the hypergeometric1 diffusions with heavy-tailed Pearson type invariant distribution, to be called Kolmogorov-Pearson diffusions (Reciprocal (inverse) gamma, Fisher-Snedecor and skew-Student diffusions). As opposed to the "classic" hypergeometric diffusions (Ornstein-Uhlebeck, Gamma/CIR, Beta/Jacobi), these diffusions have a continuous part of the spectrum, whose spectral cutoff and transition density we provide in an explicit form.

Diffusion process; Infinitesimal generator; Kolmogorov-Pearson diffusion; Sturm-Liouville equation; Transition density.

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