Unicit\'e trajectorielle des \'equations diff\'erentielles stochastiques avec temps local et temps de s\'ejour au bord

TL;DR

This paper investigates the path-wise uniqueness of solutions to a class of stochastic differential equations involving local time and boundary sojourn time, using martingale problems to establish uniqueness in law and the supremum property.

Contribution

It introduces a novel approach to prove path-wise uniqueness for SDEs with boundary local and sojourn times, extending existing theories.

Findings

Proves uniqueness in law for the class of SDEs considered.

Establishes that the supremum of two solutions is also a solution.

Demonstrates path-wise uniqueness using martingale problem techniques.

Abstract

English version of the abstract. We study path-wise uniqueness property of a class of stochastic differential equations with local time and sojourn time in the boundary. ----- French version of the abstract. Nous \'etudions l'unicit\'e trajectorielle des solutions d'une classe d'\'equations diff\'erentielles stochastiques avec temps local et temps de s\'ejour au bord. Nous utilisons le probl\'eme des martingales associ\'e pour montrer qu'il y a unicit\'e en loi, puis nous \'etablissons que le supremum de deux solutions est encore une solution.