Planning of experiments for a nonautonomous Ornstein-Uhlenbeck process (Q2913227)

The author considers nonautonomous Ornstein-Uhlenbeck processes \(X_t\), i.e.,\ solutions to stochastic differential equations of the form NEWLINE\[NEWLINE\begin{aligned} {\operatorname {d}}X_t = &\; \kappa (\bar {X} - X_t)\, {\operatorname {d}}t + \sigma (t) {\operatorname {d}}W_t, \tag{1}\\ X_0 = &\; x_0\,. \end{aligned} NEWLINE\]NEWLINEHere, the mean-reversion speed \(\kappa \) and the deterministic volatility function \(\sigma \) are known and the initial datum \(x_0\) and the asymptotic expectation \(\bar {X}\) are to be estimated based on finitely many observations in a given interval \([T_*, T^*]\). Picking sample points \(t_1 ,\ldots, t_n\), the explicit solution of the above equation gives rise to a linear regression model so that \(x_0\) and \(\bar {X}\) can be estimated in a standard way. NEWLINENEWLINEThe main objective of this article is the optimal choice of the sample points \(t_1, \ldots, t_n\). To that end, the author proves the existence of such an optimal choice for fairly general optimality criteria as well as some properties of this optimal choice. A numerical example illustrates the results.