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.
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).
Peter K. Friz. Jim Gatheral. Radoš Radoičić. "Forests, cumulants, martingales." Ann. Probab. 50 (4) 1418 - 1445, July 2022. https://doi.org/10.1214/21-AOP1560