An extension of Pratelli’s inequality

Abstract

Let X, Y be càdlàg martingales and let Y# denote the sharp function of Y. The paper contains the proof of the estimate ‖∫0∞|d〈X,Y〉t|‖1≤2‖〈X〉1∕2Y#‖1 for the total variation between X and Y. The constant 2 is shown to be the best possible. The proof rests on the construction of an appropriate special function, enjoying certain size and concavity requirements.