Path-wise study of singular diffusions.
Alexis Anagnostakis · HAL (Le Centre pour la Communication Scientifique Directe) · 2022
The main object of this thesis is the study of singular diffusion processes with a focus on sticky diffusions.Sticky diffusions were first introduced by Feller in the fifties as a case of boundary condition that can arise in the analytic description of a diffusion. Their paths spend positive amount of time at points of the state-space, giving them the appearance to 'stick' on these points. When such points are located at an attainable boundary of the state-space of the process, we call it sticky reflection.The first contribution of this thesis is to provide a suitable approximation of the local time of a sticky Itô diffusion, with statistical applications in view. We define the notion of sticky Itô diffusion and prove their path-wise descriptions. We prove that the local time of the sticky Brownian motion can be approximated by a class of high-frequency path functionals. We use the path-wise characterization to extend the result to non-explosive Itô diffusions. We prove the consistency of a stickiness estimator based on the local time approximation. We give numerical results on the stickiness estimation of a sticky Brownian motion.The second contribution of this thesis is an approximation in law of any one-dimensional diffusion by a grid-valued conditional moment-matching random walk. The convergence occurs as the maximal grid step goes to 0. We call this type of approximation 'Space-Time Markov Chain Approximation' or 'STMCA'. We also show how one can achieve optimal convergence rate by suitable choices of grids. We call 'grid tuning' the process of computing such a grid. One can use STMCAs to set up approximation schemes for any one-dimensional diffusion process. We give various illustrated approximations examples of diffusions even in the presence of sticky behavior, discontinuous or degenerate coefficients.