Estimation of the maximum value of a signal in Gaussian white noise (Q792718)

In order to study the mechanisms of nonparametric estimation the authors discuss the model of observations of the signal \(X_{\epsilon}(t)\) that has Itô stochastic differential equal to \[ dX_{\epsilon}(t)=S(t)dt+\epsilon dw(t),\quad t\in [0,1], \] where S(t) is an unknown signal, w(t) is a standard Wiener process and \(\epsilon\) is a small positive parameter. An estimator of the value max S(t) is sought. As an estimate the value \(\hat M_{\epsilon}=\sup_{U}T_{\epsilon}(t)\) is taken, where U is an open set, \(U\subset [0,1]\), and \[ T_{\epsilon}(t)= (1/\phi(\epsilon)) \int^{1}_{0} g((u-t)/\phi(\epsilon))dX_{\epsilon}(u). \] The infinitely differentiable finite function g(x) satisfies the conditions: \[ \int^{\infty}_{-\infty}x^ jg(x)dx=\delta_{j0},\quad j=0,1,...,m(\beta),\quad \beta>0, \] \(m(\beta)=[\beta]\) if \(\beta>[\beta]\), and \(=\beta -1\) if \(\beta =[\beta]\), \(\phi(\epsilon)=\chi(\epsilon^ 2\cdot \ln(1/\epsilon))^{1/(2\beta +1)}.\) Asymptotical properties of the estimate introduced are discussed in the paper.