Adaptive nonparametric drift estimation for multivariate jump diffusions under sup-norm risk

TL;DR

This paper develops adaptive nonparametric estimators for the drift function in multivariate jump diffusions, achieving optimal convergence rates under sup-norm loss with data-driven methods and novel theoretical bounds.

Contribution

It introduces two adaptive Nadaraya--Watson type estimators for jump diffusions that attain classical nonparametric convergence rates under anisotropic smoothness assumptions.

Findings

Estimators achieve optimal convergence rates in sup-norm loss.

Fully data-driven estimators perform comparably to theoretical ones.

Novel uniform moment bounds for empirical processes are established.

Abstract

We investigate nonparametric drift estimation for multidimensional jump diffusions based on continuous observations. The results are derived under anisotropic smoothness assumptions and the estimators' performance is measured in terms of the sup-norm loss. We present two different Nadaraya--Watson type estimators, which are both shown to achieve the classical nonparametric rate of convergence under varying assumptions on the jump measure. Fully data-driven versions of both estimators are also introduced and shown to attain the same rate of convergence. The results rely on novel uniform moment bounds for empirical processes associated to the investigated jump diffusion, which are of independent interest.