Sanz, Marta Local time for two-parameter continuous martingales with respect to the quadratic variation. (English) Zbl 0645.60066 Ann. Probab. 16, No. 2, 778-792 (1988). The author studies the local time for two-parameter continuous martingales M as a density of the “measure of sojourn time” with respect to the quadratic variation \(<M>\). First she shows that there exists a process \(\{L(x,s,t);\) \(x\in {\mathbb{R}}\setminus \{0\},\) \((s,t)\in {\mathbb{R}}^ 2_+\}\) satisfying the occupation density formula and which is a.s. jointly continuous and, for fixed \(x\neq 0\), continuously differentiable in s and t. Then she gives the modules of continuity of \(L(\cdot,s,t)\) under further assumptions on \(<M>\). Finally she applies the results to the martingales with respect to the filtration generated by the Brownian sheet and to martingales with respect to the product filtration generated by independent multidimensional Brownian motions. The examples show in particular that \(L(0,s,t)\) may be infinite a.s. The results generalize those of J. B. Walsh in: Temps locaux. Exposés du séminaire J. Azema - M. Yor (1976-1977), Astérisque 52-53 (1978; Zbl 0385.60063). Reviewer: M.Dozzi Cited in 1 ReviewCited in 2 Documents MSC: 60H99 Stochastic analysis 60G17 Sample path properties 60G44 Martingales with continuous parameter 60J55 Local time and additive functionals Keywords:local time for two-parameter continuous martingales; quadratic variation; occupation density formula; modules of continuity; Brownian sheet Citations:Zbl 0385.60063 PDF BibTeX XML Cite \textit{M. Sanz}, Ann. Probab. 16, No. 2, 778--792 (1988; Zbl 0645.60066) Full Text: DOI OpenURL