Realized cumulants for martingales

TL;DR

This paper introduces realized cumulants for martingales using the aggregation property, establishing a link with Bell polynomials, and extends existing cumulant recursion formulas for financial modeling.

Contribution

It generalizes realized cumulants for martingales, connecting the aggregation property with Bell polynomials and extending recent cumulant recursion results.

Findings

Established a relation between aggregation property and Bell polynomials.

Provided an alternative proof for a cumulant recursion formula.

Extended cumulant recursion formulas for martingale analysis.

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).