Invariant density estimation for multidimensional diffusions

We consider an Rd dimensional homogeneous diffusion process with a unique invariant density f. We construct a kernel type estimator for the invariant density and study its mean-square convergence. We find that this estimator reaches in a specific minimax sense a rate that is slower than parametric but faster than in classical d-dimensional estimation problems. Finally we examine the almost sure (pointwise and uniform) behavior of the estimator and we give examples. © 2007 Springer.