Sparse inference of the drift of a high-dimensional Ornstein-Uhlenbeck process. (English) Zbl 1404.60049
Summary: Given the observation of a high-dimensional Ornstein-Uhlenbeck (OU) process in continuous time, we are interested in inference on the drift parameter under a row-sparsity assumption. Towards that aim, we consider the negative log-likelihood of the process, penalized by an \(\ell^1\)-penalization (Lasso and Adaptive Lasso). We provide both finite- and large-sample results for this procedure, by means of a sharp oracle inequality, and a limit theorem in the long-time asymptotics, including asymptotic consistency for variable selection. As a by-product, we point out the fact that for the Ornstein-Uhlenbeck process, one does not need an assumption of restricted eigenvalue type in order to derive fast rates for the Lasso, while it is well-known to be mandatory for linear regression for instance. Numerical results illustrate the benefits of this penalized procedure compared to standard maximum likelihood approaches both on simulations and real-world financial data.
MSC:
| 60G15 | Gaussian processes |
| 62H12 | Estimation in multivariate analysis |
| 62M99 | Inference from stochastic processes |
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