The continuity of the quadratic variation of two-parameter martingales (Q1112447)

Let \(M=(M_ t;t\in {\mathbb{R}}^ 2_+)\) be an L \(log^+L\)-integrable two- parameter martingale. According to a theorem by \textit{D. Bakry} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 149-157 (1979; Zbl 0419.60051)] and by \textit{A. Millet} and \textit{L. Sucheston} [ibid. 56, 21-45 (1981; Zbl 0452.60058)] M has a version whose trajectories are continuous for approach from the right upper quadrant and possess limits for approach from the remaining three quadrants. In this paper the authors present results on the continuity of the quadratic variation [M] of M. They show that [M] is continuous resp. has no jump points resp. has at most jumps in the direction of the \(i^{th}\) coordinate axis \((i=1,2)\) if M has the same property. This work is a continuation of the results published by \textit{P. Imkeller} [Two-parameter martingales and their quadratic variation. Lecture Notes Math. 1308 (1988; Zbl 0656.60056)] and by \textit{D. Nualart} [Ann. Probab. 12, 445-457 (1984; Zbl 0538.60049)].