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On lower and upper functions for square integrable martingales. (English) Zbl 1032.60040

Proc. Steklov Inst. Math. 237, 281-292 (2002) and Tr. Mat. Inst. Steklova 237, 290-301 (2002).
Let \(M= (M_t,{\mathcal F}_t)\) be a locally square integrable martingale such that \(\langle M\rangle_t\to \infty\) a.s. (as \(t\to\infty\)) and \(|\Delta M_s|\leq g(\langle M\rangle_s)\) for \(s\geq t_0> 0\) where \(g\) is a nonnegative nondecreasing continuous function (\(\langle M\rangle\) denoting the predictable quadratic characteristic of \(M\)). Let \(\varphi\) denote a nonnegative nondecreasing continuous function. The authors obtain a sufficient condition (similar to the Kolmogorov-Petrovskij test) for \(\varphi(\langle M\rangle)\) to be a lower function for \(|M|\). In particular, in the case where \(\varphi(t)= \sqrt{2t\ln\ln t}\) and \(g(t)= O(t^{1/2}/(\ln t)^{1+\delta})\) it turns out that \(\sqrt{2\langle M\rangle_t\ln\ln\langle M\rangle_t}\) is a lower function for \(|M|\).
For the entire collection see [Zbl 1007.00020].

MSC:

60G44 Martingales with continuous parameter
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