The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

Abstract

Abstract. The Pearson diffusions form a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well‐known instances are the Ornstein–Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy‐tailed and skew marginals are included. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. Explicit optimal martingale estimating functions are found. The discussion covers GMM, quasi‐likelihood, non‐linear weighted least squares estimation and likelihood inference too. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions and Pearson stochastic volatility models. For the non‐Markov models, explicit optimal prediction‐based estimating functions are found. The estimators are shown to be consistent and asymptotically normal.