arXiv subjects
Publications and source records attributed to Ashkan Nikeghbali.
This text is a survey of the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which were developed by the French probability school and which are usually written in French. The material presented in the last three chapters is less standard and takes into account some recent developments.
This note deals with the question: what remains of the Burkholder-Davis-Gundy inequalities when stopping times $T$ are replaced by arbitrary random times $ρ$? We prove that these inequalities still hold when $T$ is a pseudo-stopping time and never holds for ends of predictable sets.
Recently, D. Williams \cite{williams} gave an explicit example of a random time $ρ$ associated with Brownian motion such that $ρ$ is not a stopping time but $\mathbb{E}M_ρ=\mathbb{E}M_{0}$ for every bounded martingale $M$. The aim of this paper is to give some characterizations for such random times, which we call pseudo-stopping times, and to construct further examples, using techniques of progressive enlargements of filtrations.