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 Tr. Mat. Inst. Steklova, 2002, Volume 237, Pages 290–301 (Mi tm340)

On Lower and Upper Functions for Square Integrable Martingales

A. N. Shiryaeva, E. Valkeilab, L. Yu. Vostrikovac

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Helsinki
c Université d'Angers

Abstract: We consider a locally square integrable martingale $M = (M_t,\mathcal F_t)_{t\ge 0}$ satisfying $\lim _{t\to\infty }\langle M\rangle _t = +\infty$ ($\mathsf P$-a.s.), with predictably bounded jumps $|\Delta M_s| \le g(\langle M\rangle_s)$ for $s\ge t_0\ge 0$, where $g$ is a nonnegative nondecreasing continuous function and $\langle M \rangle$ is the predictable quadratic characteristic of $M$. For a nonnegative nondecreasing continuous function $\phi$, we give a sufficient condition similar to the Kolmogorov–Petrovskii test saying when $\phi (\langle M\rangle)$ is a lower function for $|M|$. In particular, if $\phi(t)=\sqrt{2t\ln\ln t}$ and $g(t)=O({t^{1/2}}/{ (\ln t)^{1+\delta}})$, we obtain that $\sqrt {2\langle M\rangle \ln\ln \langle M\rangle _t}$ is lower for $|M|$ and $\limsup _{t\to \infty} {|M_t|}/ {\sqrt { 2\langle M\rangle \ln\ln \langle M\rangle _t}}\ge 1$ $\mathsf P$-a.s. If the predictable quadratic characteristic $\langle M\rangle$ is continuous in $t$, then, under some supplementary conditions on jumps of $M$, we prove an analogous result for $\phi (t) = \sqrt {2t\ln\ln t}$ and $g(t)=O (t^{1/2}/(\ln \ln t)^{3/2})$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 237, 281–292

Bibliographic databases:
UDC: 519.2+519.8
Received in January 2002
Language:

Citation: A. N. Shiryaev, E. Valkeila, L. Yu. Vostrikova, “On Lower and Upper Functions for Square Integrable Martingales”, Stochastic financial mathematics, Collected papers, Tr. Mat. Inst. Steklova, 237, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 290–301; Proc. Steklov Inst. Math., 237 (2002), 281–292

Citation in format AMSBIB
\Bibitem{ShiValVos02} \by A.~N.~Shiryaev, E.~Valkeila, L.~Yu.~Vostrikova \paper On Lower and Upper Functions for Square Integrable Martingales \inbook Stochastic financial mathematics \bookinfo Collected papers \serial Tr. Mat. Inst. Steklova \yr 2002 \vol 237 \pages 290--301 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm340} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1976524} \zmath{https://zbmath.org/?q=an:1032.60040} \transl \jour Proc. Steklov Inst. Math. \yr 2002 \vol 237 \pages 281--292 

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This publication is cited in the following articles:
1. Lepski O., “Upper functions for $\mathbb{L}_{p}$-norms of Gaussian random fields”, Bernoulli, 22:2 (2016), 732–773
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