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

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 minimax optimal 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.