Forests, cumulants, martingales

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