Description du sujet de thèse

Stochastic differential equations (SDEs) play a fundamental role in applied mathematics, modeling various real-world phenomena in fields such as finance, physics, biology, or the social sciences. A particularly important class of SDEs incorporates the mean state of the system, leading to distribution-dependent dynamics (also known as McKean-Vlasov dynamics). The study of such SDEs, as well as their approximations by interacting particle systems (IPS), has recently seen renewed interest thanks to advances in propagation of chaos techniques.

While the theoretical and numerical aspects of distribution-dependent SDEs have been extensively studied, their statistical analysis remains a relatively recent field. Even in the classical setting of SDEs driven by Brownian motion, many statistical contributions have only emerged during the 2020s. The need for robust statistical tools becomes even more pressing when considering SDEs driven by fractional Brownian motion (fBm), which introduces memory effects and requires methodologies that go beyond the classical Itô calculus framework. In this area, the work Amorino, C., Nourdin, I., Shevchenko, R., Fractional Interacting Particle System: Drift Parameter Estimation Via Malliavin Calculus (preprint), which studies drift estimators with linear dependence on the parameter based on continuous observations, constitutes a pioneering contribution.

Contexte de travail

Building on the existing literature in the non-interacting setting, we aim to construct drift estimators in various contexts, including going beyond the linear case, and also considering multiplicative noise. The thesis includes a visit of several months with Grigorios Pavliotis at the IRL Abraham De Moivre in London.