Timing Observations of Diffusions

TL;DR

This paper develops an adaptive observation timing policy for Itô diffusions that maximizes Fisher information, significantly improving parameter estimation accuracy over uniform sampling.

Contribution

It introduces a novel adaptive sampling policy for diffusions that optimally allocates observation times to enhance parameter estimation.

Findings

Reduces estimation variance by up to 75% compared to uniform sampling.

Policy adapts based on current parameter estimates and prior information.

Numerical results demonstrate substantial efficiency gains.

Abstract

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 $\theta$.