Estimation et tests dans les processus de diffusion à dérive non régulière

Abstract

Many natural, economic, and biological phenomena involve random fluctuations that cannot be captured by purely deterministic models. Stochastic differential equations (SDEs) provide a suitable framework by combining drift and diffusion components. This thesis focuses on a diffusion process with proportional delay and non-regular drift. Two main objectives are addressed: estimating the delay parameter via maximum likelihood in the small-diffusion regime, and constructing a simple versus simple parametric test based on the likelihood ratio. Theoretical results, inspired by the works of Ibragimov, Has’minskii, and Kutoyants, are supported by numerical simulations.