Quasi-sure product variation of two-parameter smooth martingales on the Wiener space (Q2508632)

The Doob-Meyer decomposition theorem has been extended to two-parameter martingales \((M_z)_{z\in T}\), for which it ensures the existence and uniqueness of a quadratic variation, i.e., a predictable, increasing process \((\langle M\rangle_z )_{z\in T}\) vanishing on the axes and such that \(M^2-\langle M\rangle\) is a weak martingale. In addition to the quadratic variation, the two-parameter case gives rise to a notion of product variation process which has been studied by several authors, due to its role in establishing the two-parameter Itô and Tanaka formulas. In this paper, using the Malliavin calculus on abstract Wiener spaces, the authors prove that the finite sums defining the product variation of two-parameter martingales converge in the quasi-sure sense. They also prove a similar result relative to the quadratic variation and covariation processes.