Nonparametric Identification for Diffusion Processes

It is proved that under a specific condition (so-called condition $G_2 $) on the transition probability operator of a measurable stationary Markov process, a recursive kernel estimate of the initial density is convergent in quadratic mean. Assumptions on the differential stochastic equations driven by Brownian motion are derived under which the stationary solution satisfies condition $G_2 $. The above results are applied to solve a class of nonlinear identification problems.