Timing observations of diffusions

Abstract

This paper addresses a problem in experimental design: We consider Itô diffusions specified by some θ ∈ R and assume that we are allowed to observe their sample paths only n times before a terminal time τ < ∞. We propose a policy for timing these observations to optimally estimate θ. Our policy is adaptive (meaning it leverages earlier observations), and it maximizes the expected Fisher information for θ 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 θ.