Quadratic Variation for a Class of $L \log^+ L$-Bounded Two-parameter Martingales

Abstract

Let $M = (M_t)_{t\in\lbrack 0,1\rbrack^2}$ be a two-parameter $L\log^+ L$-bounded (not necessarily continuous) martingale. Assume that the marginal filtrations in the first and second directions are quasi-left continuous. We prove the existence of quadratic variation in the sense of convergence in probability. This is done first for bounded martingales. The extension to the general case is obtained by approximating a given martingale by its bounded truncations and using a two-parameter version of the square function inequality of Burkholder.