Open Access
July 2022 Forests, cumulants, martingales
Peter K. Friz, Jim Gatheral, Radoš Radoičić
Author Affiliations +
Ann. Probab. 50(4): 1418-1445 (July 2022). DOI: 10.1214/21-AOP1560


This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability space. We establish a “broken exponential martingale” expansion that generalizes and unifies the exponentiation result of Alòs, Gatheral, and Radoičić (SSRN’17; Quant. Finance 20 (2020) 13–27) and the cumulant recursion formula of Lacoin, Rhodes, and Vargas (arXiv; (2019)). Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula, as required by many of our examples, including iterated stochastic integrals, Lévy area, Bessel processes, KPZ with smooth noise, Wiener–Itô chaos, and “rough” stochastic (forward) variance models.

Funding Statement

PKF has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 683164), DFG Research Unit FOR2402 (TP 2) and DFG Excellence Cluster MATH+ (AA4-2).


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Peter K. Friz. Jim Gatheral. Radoš Radoičić. "Forests, cumulants, martingales." Ann. Probab. 50 (4) 1418 - 1445, July 2022.


Received: 1 July 2020; Revised: 1 July 2021; Published: July 2022
First available in Project Euclid: 11 May 2022

MathSciNet: MR4420423
zbMATH: 07527829
Digital Object Identifier: 10.1214/21-AOP1560

Primary: 60G44 , 60H99
Secondary: 60L70

Keywords: continuous martingales , Cumulants , diamond product , forests , Hermite polynomials , Heston and forward variance models , KPZ type (Wild) expansion , Lévy area , moments , regular perturbation , trees , trees , Wiener Chaos

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.50 • No. 4 • July 2022
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