On cusp estimation of ergodic diffusion process

The properties of the maximum-likelihood (MLE) and Bayesian (BE) estimators of the parameter of ergodic diffusion process are studied in the situation when the trend coefficient has a cusp, i.e., it admits the representation S( ϑ, x)= d( x− ϑ)| x− ϑ| p + h( x− ϑ), where p∈(0, 1 2 ) , d( x)= a for x0, and the function h(·) is regular. This problem of estimation is not regular (Fisher information is equal to infinity), and it is shown that the rate of convergence of the estimators is T 1/(2 p+1) , the estimators MLE and BE have different limit laws, and the BE is asymptotically optimal.