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A one dimensional diffusion process X = {X t , 0 ≤ t ≤ T }, with drift b(x) and diffusion coefficient σ(θ, x) = √ θσ(x) known up to θ > 0, is supposed to switch volatility regime at some point t * ∈ (0, T ). On the basis of discrete time observations from X, the problem is the one of estimating the instant of change in the volatility structure t * as well as the two values of θ, say θ 1 and θ 2 , before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length ∆ n with n∆ n = T . To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant.

The least absolute shrinkage and selection operator (LASSO) is a widely used statistical methodology for simultaneous estimation and variable selection. It is a shrinkage estimation method that allows one to select parsimonious models. In other words, this method estimates the redundant parameters as zero in the large samples and reduces variance of estimates. In recent years, many authors analyzed this technique from a theoretical and applied point of view. We introduce and study the adaptive LASSO problem for discretely observed multivariate diffusion processes. We prove oracle properties and also derive the asymptotic distribution of the LASSO estimator. This is a nontrivial extension of previous results by Wang and Leng (2007, Journal of the American Statistical Association, 102(479), 1039-1048) on LASSO estimation because of different rates of convergence of the estimators in the drift and diffusion coefficients. We perform simulations and real data analysis to provide some evidence on the applicability of this method.

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