Forests, Cumulants, Martingales
25 Pages Posted: 30 Jun 2020
Date Written: June 4, 2020
This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a “broken exponential martingale” expansion that generalizes and unifies the exponentiation result of Alòs, Gatheral, and Radoičić ́ (SSRN’17; [AGR20]) and the cumulant recursion formula of Lacoin, Rhodes, and Vargas (arXiv; [LRV19]). 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.
Keywords: forests, trees, continuous martingales, diamond product, cumulants, mo- ments, Hermite polynomials, regular perturbation, KPZ type (Wild) expansion, trees, Le ́vy area, Wiener chaos, Heston and forward variance models;
JEL Classification: 60G44, 60H99, 60L70
Suggested Citation: Suggested Citation