Parametric probability density estimation based on an approximation by a discretized stochastic differential equation

The authors present a parametric probability density estimation technique for Markov processes defined by a first-order nonlinear autoregressive equation. It is based on the approximation of these processes as sampled versions of the continuous-time solutions of stochastic differential equations (SDEs) via the discretization scheme presented by T. Ozaki (1985). First, a polynomial estimate of the nonlinear recursion function is obtained from the data and then a suitable transformation of its coefficients is performed in order to obtain an estimate of the function in the corresponding SDE. The PDF estimate is then the equilibrium PDF of this SDE