Stochastic analysis on Gaussian space applied to drift estimation

TL;DR

This paper develops efficient and super-efficient estimators for the drift of Gaussian processes using advanced stochastic analysis techniques, extending classical James-Stein estimators to a nonparametric Gaussian setting.

Contribution

It introduces novel estimators based on Malliavin calculus and superharmonic functionals, demonstrating their minimaxity and super-efficiency in Gaussian process drift estimation.

Findings

Constructed minimax estimators for Gaussian process drift.

Developed Stein-type super-efficient estimators using Malliavin calculus.

Validated estimators through numerical simulations.

Abstract

In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper.