Regularization methods for diffusion processes: an approach based on elastic-net estimation

This thesis develops a methodological contribution to penalized inference for stochastic processes through the construction and analysis of an Adaptive Elastic-Net estimator for sparse parametric diffusion models observed at discrete times. The problem is addressed in a high-frequency and increasing-horizon framework, where the exact likelihood is generally unavailable and the drift and diffusion parameters are governed by different convergence rates. The proposed estimator is constructed from a least-squares approximation of a likelihood-type contrast around a preliminary estimator and combines an adaptive l1 component, which induces sparsity, with a quadratic l2 component, which stabilizes the selection mechanism in the presence of correlated predictors. The thesis connects the theory of diffusion processes, sparse statistical learning and computational optimization through a single mixed-rate regularization problem. The theoretical analysis proves that the Adaptive Elastic-Net estimator recovers the sparse structure of the diffusion model while preserving the mixed-rate asymptotic behavior of the active parameters. The analysis is further extended beyond the oracle result by establishing Lr control of the rescaled estimation error and by deriving non-asymptotic bounds for the estimation and one-step-ahead prediction. Consistently with the theoretical results, the estimator is made computationally tractable through pathwise optimization methods and its finite-sample behavior is then investigated through numerical experiments with dynamically correlated predictors and on a real data application. The resulting framework provides a sparse, interpretable and computationally stable method for estimation and variable selection in discretely observed diffusion processes.