Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions

TL;DR

This paper introduces a nonparametric method for estimating the drift function of multiple correlated diffusion processes observed over a fixed time interval, addressing the challenge of unknown correlations.

Contribution

It proposes a novel drift estimator that does not require knowledge of the correlation matrix, extending nonparametric diffusion estimation to correlated processes.

Findings

Estimator's integrated mean squared risk is bounded

Adaptive procedure for the estimator is developed

Numerical experiments confirm practical effectiveness

Abstract

In the present paper, we consider that $N$ diffusion processes $X^1,\dots,X^N$ are observed on $[0,T]$, where $T$ is fixed and $N$ grows to infinity. Contrary to most of the recent works, we no longer assume that the processes are independent. The dependency is modeled through correlations between the Brownian motions driving the diffusion processes. A nonparametric estimator of the drift function, which does not use the knowledge of the correlation matrix, is proposed and studied. Its integrated mean squared risk is bounded and an adaptive procedure is proposed. Few theoretical tools to handle this kind of dependency are available, and this makes our results new. Numerical experiments show that the procedure works in practice.