Realized cumulants for martingales

Masaaki Fukasawa; Kazuki Matsushita

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arXiv · 2010.08293

Realized cumulants for martingales

Masaaki Fukasawa·Kazuki Matsushita

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Abstract

Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and Friz et al. (2020).

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Masaaki Fukasawa, Kazuki Matsushita. 2020-10-16. Realized cumulants for martingales. https://arxiv.org/abs/2010.08293

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