Prediction and Related Topics for a Scalar Linear Stochastic Equation with a Rosenblatt Process Noise

A scalar linear stochastic system with a Rosenblatt noise process is described and a prediction problem is explicitly solved for this system. Some other topics related to this stochastic system are also considered. Since the Rosenblatt processes are defined by double Wiener-Itô integrals the usual stochastic calculus for Brownian motion is not appropriate. Stochastic calculus is not used for the double integrals. Instead one uses the Wiener-ô double integral definition that has an appropriate orthogonality for expectation. For the prediction problem solution instead one uses some stochastic calculus for a double Wiener-Itô so that such integrals have expectation zero. An ergodic control for a scalar controlled system with a Rosenblatt noise is also briefly described.