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Local time for two-parameter continuous martingales with respect to the quadratic variation. (English) Zbl 0645.60066

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

MSC:

60H99 Stochastic analysis
60G17 Sample path properties
60G44 Martingales with continuous parameter
60J55 Local time and additive functionals

Citations:

Zbl 0385.60063
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