Predicting Integrals of Stochastic Processes

Abstract

Consider predicting an integral of a stochastic process based on $n$ observations of the stochastic process. Among all linear predictors, an optimal quadrature rule picks the $n$ observation locations and the weights assigned to them to minimize the mean squared error of the prediction. While optimal quadrature rules are usually unattainable, it is possible to find rules that have good asymptotic properties as $n \rightarrow \infty$. Previous work has considered processes whose local behavior is like $m$-fold integrated Brownian motion for $m$ a nonnegative integer. This paper obtains some asymptotic properties for quadrature rules based on median sampling for processes whose local behavior is not like $m$-fold integrated Brownian motion for any $m$.