Concentration Inequalities for Martingales with Jumps

Over the last three decades, there have been a renewed interest in the area of exponential concentration inequalities in probability theory and mathematical statistics. It has the origin from the classical deviation inequalities of sums of independent random variables like Bernstein, Bennett, and Hoeffding. Furthermore, exponential inequalities for discrete time martingales have been also studied by many authors. In this talk, I will present our recent work on concentration Inequalities for continuous time local martingales. We obtain the classical Bernstein type inequality, de la Pena's inequality and exponential concentration inequality for local martingales with jumps under the exponential moments or bounded jumps assumption. Besides, we consider the continuous time matrix-valued local martingales and obtain a refined concentration inequality for norms of matrix operators through a new exponential supermartingale for traces.