Sparse inference of the drift of a high-dimensional Ornstein-Uhlenbeck process
This work addresses the challenge of parameter estimation in high-dimensional stochastic processes for applications like financial modeling, but it is incremental as it extends existing penalized methods to a specific continuous-time setting.
The paper tackles the problem of inferring the drift parameter of a high-dimensional Ornstein-Uhlenbeck process under row-sparsity, using Lasso and Adaptive Lasso penalization. It provides non-asymptotic and asymptotic results, including sharp oracle inequalities and asymptotic consistency for variable selection, with numerical benefits shown on simulations and real-world financial data.
Given the observation of a high-dimensional Ornstein-Uhlenbeck (OU) process in continuous time, we proceed to the inference of 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 non-asymptotic and asymptotic 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.