Timing Observations of Diffusions

This paper addresses a problem in experimental design: We consider It\^o diffusions specified by some $\theta \in \mathbb{R}$ and assume that we are allowed to observe their sample paths only $n$ times before a terminal time $\tau < \infty$. We propose a policy for timing these observations to optimally estimate $\theta$. Our policy is adaptive (meaning it leverages earlier observations), and it maximizes the expected Fisher information for $\theta$ carried by the observations. In numerical studies, this design reduces the variation of estimated parameters by as much as 75% relative to observations spaced uniformly in time. The policy depends on the value of the parameter being estimated, so we also discuss strategies for incorporating Bayesian priors over $\theta$.