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The Pearson family of ergodic diffusions with a quadratic diffusioncoefficient and a linear force are characterized by explicit dynamics of theirinteger moments and by explicit relaxation spectral properties towards theirsteady state. Besides the Ornstein-Uhlenbeck process with a Gaussian steadystate, the other representative examples of the Pearson family are theSquare-Root or the Cox-Ingersoll-Ross process converging towards theGamma-distribution, the Jacobi process converging towards theBeta-distribution, the reciprocal-Gamma process (corresponding to anexponential functional of the Brownian motion) that converges towards theInverse-Gamma-distribution, the Fisher-Snedecor process, and the Studentprocess, so that the last three steady states display heavy-tails. The goal ofthe present paper is to analyze the large deviations properties of thesevarious diffusion processes in a unified framework. We first consider the Level1 concerning time-averaged observables over a large time-window $T$ : we writethe first rescaled cumulants for generic observables and we identify thespecific observables whose large deviations can be explicitly computed from thedominant eigenvalue of the appropriate deformed-generator. The explicit largedeviations at Level 2 concerning the time-averaged density are then used toanalyze the statistical inference of model parameters from data on a very longstochastic trajectory in order to obtain the explicit rate function for the twoinferred parameters of the Pearson linear force.

Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force