Penalized nonparametric drift estimation for a multidimensional diffusion process

Abstract

We consider a multi-dimensional diffusion process (X t ) t≥0 with drift vector b and diffusion matrix Σ. This process is observed at n+1 discrete times with regular sampling interval Δ. We review sufficient conditions for the existence and uniqueness of an invariant density. In a second step, we assume that the process is stationary, and estimate the drift function b on a compact set K in a nonparametric way. For this purpose, we consider a family of finite dimensional linear subspaces of L 2(K), and compute a collection of drift estimators on every subspace by a penalized least-squares approach. We introduce a penalty function and select the best drift estimator. We obtain a bound for the risk of the resulting adaptive estimator. Our method fits for any dimension d, but, for safe of clarity, we focus on the case d=2. We also provide several examples of two-dimensional diffusions satisfying our assumptions, and realize various simulations. Our results illustrate the theoretical properties of our estimators.