Optimal Measurement Times for a Small Number of Measures of a Brownian Motion over a Finite Period

The measure timetable plays a critical role for the accuracy of the estimator. This article deals with the optimization of the schedule of measures for observing a random process in time using a Kalman filter, when the length of the process is finite and fixed, and a fixed number of measures are available. The measuring devices are allowed to differ. The mean variance of the estimator is chosen as criterion for optimality. The cases of $1$ or $2$ measures are studied in detail, and analytical formulas are provided.