Penalized nonparametric mean square estimation of the coefficients of diffusion processes

Abstract

We consider a one-dimensional diffusion process (Xt) which is observed at n + 1 discrete times with regular sampling interval ∆. Assuming that (Xt) is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2007, Vol. 13, No. 2, 514-543. This reprint differs from the original in pagination and typographic detail.

Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process

“…The proof follows the lines of Comte et al [5]. Recall that from the definition of γ t it follows that on A t ,…”

“…It is well known, see for instance Comte et al [5], that for this model the assumption of norm connection 2.4.2.4 is satisfied. Note moreover that for a fixed ϕ j,l ∈ S t ,…”

“…We consider the space of piecewise polynomials, as introduced for example in Baraud et al [2,3] and Comte et al [5].…”

“…Firstly, we aim at introducing a nonparametric estimation procedure based on model selection. Our estimator is obtained by minimizing a contrast function within a fixed finite-dimensional linear sub-space of L 2 (K, dx) -quite in the spirit of mean square estimation and following ideas presented by Comte et al [5], for discretely observed diffusions. These finite-dimensional sub-spaces include spaces such as piecewise polynomials or compactly supported wavelets.…”

Variable bandwidth local maximum likelihood type estimation for diffusion processes

“…Reference [2] improved estimation of drift parameters of diffusion processes for interest rates by incorporating information in bond prices. So, recently in the literature, the statistical inference for diffusion processes based on discrete observations has often been of concern; for example, see [3][4][5][6] and its references for parametric estimation, see [7][8][9] and the references therein for a semi-parametric estimation and see [10][11][12][13][14][15][16] and the references therein for a nonparametric estimation. As is well known, the first to consider nonparametric estimation for the diffusion coefficient in model (1) with discrete-time observation was [17], where a kernel type estimator was considered.…”