Asymmetric Doob inequalities in continuous time

The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let (M,τ) be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras (Mt)0≤t≤1 whose union is weak-⁎ dense in M. Let Et denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for 1