  We consider the problem of parameter estimation in the case of observation of
the trajectory of diffusion process. We suppose that the drift coefficient has
a singularity of cusp-type and the unknown parameter corresponds to the
position of the point of the cusp. The asymptotic properties of the maximum
likelihood estimator and Bayesian estimators are described in the asymptotics
of {\it small noise}, i.e., as the diffusion coefficient tends to zero. The
consistency, limit distributions and the convergence of moments of these
estimators are established.
