A 1-dimensional nonlinear filtering problem

We will concentrate on a 1-dimensional nonlinear filtering problem, which allows an explicit solution in terms of a stochastic differential equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat X_t $$\end{document}.