Oracle inequalities for the stochastic differential equations

TL;DR

This paper surveys recent advances in adaptive robust nonparametric estimation methods for continuous-time regression models with complex noises, including Levy, Ornstein-Uhlenbeck, and semi-Markov processes, emphasizing oracle inequalities and model selection improvements.

Contribution

It provides a comprehensive overview of sharp oracle inequalities and adaptive model selection techniques for nonparametric estimation under complex stochastic noises.

Findings

Representation of general model selection methods.

Development of sharp oracle inequalities for robust estimation.

Recent improvements in model selection methods.

Abstract

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi - martingale noises with jumps. The noises are modeled by the L\'evy processes, the Ornstein -- Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.