Forests, cumulants, martingales
Abstract
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 Al{\`o}s, Gatheral, and Radoi\v{c}i\'c and the cumulant recursion formula of Lacoin, Rhodes, and Vargas. 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, L\'evy area, Bessel processes, KPZ with smooth noise, Wiener-It\^o chaos and "rough" stochastic (forward) variance models.
Keywords
Cite
@article{arxiv.2002.01448,
title = {Forests, cumulants, martingales},
author = {Peter K. Friz and Jim Gatheral and Radoš Radoičić},
journal= {arXiv preprint arXiv:2002.01448},
year = {2020}
}
Comments
25 pages. Additional applications. Analytic proof of forest reordering. Relation to Al\`os et al. (SSRN'17, Quant. Finance '20) and Lacoin et al. (arXiv'19) discussed in detail