论文标题
关于某些Martingale不平等的注释
A Note on Some Martingale Inequalities
论文作者
论文摘要
我们得出了与著名的伯克霍尔德不平等现象相似的时间散落和时间连续的群众的不平等现象。对于$ l^p（ω）$的时间散落案例，请使用$ l^p（ω）$的任意群众，而在Itôintectals定义的时间连续案例中，W.R.T.考虑了多维维纳过程。时间差异的群众的估计与I. Pinelis（1994）的更一般结果有关，并被这种特殊环境的不同，更基本的证据证明是敏锐的。此外，对于时间连续的群众，提出的不平等是M. Zakai（1967）和E. Rio（2009）证明了类似估计值的概括。尤其是，与原始伯克持有人不等式相比，这些不平等具有较小的常数。因此，所提出的不平等值在例如随机分析和随机数字中具有很高的价值。
We derive inequalities for time-discrete and time-continuous martingales that are similar to the well-known Burkholder inequalities. For the time-discrete case arbitrary martingales in $L^p(Ω)$ are treated, whereas in the time-continuous case martingales defined by Itô integrals w.r.t. a multi-dimensional Wiener process are considered. The estimates for the time-discrete martingales are related to the more general results by I. Pinelis (1994) and are proved to be sharp by a different and more elementary proof for this special setting. Further, for time-continuous martingales the presented inequalities are generalizations of similar estimates proved by M. Zakai (1967) and E. Rio (2009) to the general multi-dimensional case. Especially, these inequalities possess smaller constants compared to the ones that result if the original Burkholder inequalities would be applied for such estimates. Therefore, the presented inequalities are highly valuable in, e.g., stochastic analysis and stochastic numerics.
