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On the law of the maximum and the local time of a uniformly integrable continuous martingale. (Sur la loi du maximum et du temps local d’une martingale continue uniformément intégrable.) (French) Zbl 0807.60041

The author characterizes the laws of the maximum (respectively the local time in zero) of uniformly integrable and continuous martingales started at zero. More precisely if \({\mathcal S}_ c\) denotes the set of laws of this class of martingales, if \(\nu\) is a probability law on \(\mathbb{R}_ +\), \(a\) the upper bound of the support of \(\nu\), \(\nu = \nu^{(c)} + \nu^{(s)}\), \(\nu^{(c)} (dx) = \rho(x) dx\), the Lebesgue’s decomposition of \(\nu\), then \(\nu \in {\mathcal S}_ c\) if and only if
(i) \(\rho(x) > 0 \quad \text{on}\quad [0,a) \text{ a.e.}\),
(ii) \(\lim_{x \to +\infty} (x\nu [x,+\infty)) = 0\),
(iii) \(\int^ \infty_ 0 | x\rho(x) - \rho[x, +\infty) | dx + \int^ \infty_ 0 td\nu^{(s)} (t) < \infty.\)
The second part of this work gives a refinement of the maximal inequality [L. E. Dubins and D. Gilat, Proc. Am. Math. Soc. 68, 337-338 (1978; Zbl 0351.60049)].

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
60G46 Martingales and classical analysis
60J55 Local time and additive functionals
60J65 Brownian motion
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