Martingale Inequalities for the Maximum via Pathwise Arguments
Martingale Inequalities for the Maximum via Pathwise Arguments
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通过路径论证求最大值的鞅不等式
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
N. Touzi
中科院分区：
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作者：
J. Obłój;Peter Spoida;N. Touzi
研究了一类涉及运行极大过程的鞅不等式。它们是从Henry-Schmiddere等人（Ann. Appl. Probab.，2015 [arxiv：1203.6877v3]），并提供n个中间时间点的边际分布方面的运行最大值函数的期望上限。类的不平等是丰富的，我们表明，在一般情况下，没有不等式是一致尖锐的任何两个不等式，我们指定鞅，使一个或另一个不等式是尖锐的。我们使用我们的不等式恢复Doob的LP不等式。进一步，当p = 1时，我们改进了已知的不等式，当p < 1时，我们得到了新的不等式。
We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry-Labordere et al. (Ann. Appl. Probab., 2015 [arxiv:1203.6877v3]) and provide an upper bound on the expectation of a function of the running maximum in terms of marginal distributions at n intermediate time points. The class of inequalities is rich and we show that in general no inequality is uniformly sharp—for any two inequalities we specify martingales such that one or the other inequality is sharper. We use our inequalities to recover Doob’s L p inequalities. Further, for p = 1 we refine the known inequality and for p < 1 we obtain new inequalities.