Some norm estimates for semimartingales

In this paper we introduce a new type of norm for semimartingales. Our norm is defined in the spirit of quasimartingales, and it characterizes square integrable semimartingales. This work is motivated by our study of zero-sum stochastic differential games, whose value process we conjecture to be a semimartingale under a class of probability measures under some conditions. The norm introduced here seems to be the right one to study general square integrable semimartingales, and it is also suitable for studying semimartingales under nonlinear expectation. Using a similar idea, we introduce a new norm for the barriers of doubly reflected BSDEs and establish some a priori estimates for the solutions. Our norm provides an alternative but more tractable characterization for the standard Mokobodski's condition in the literature.