Forests, Cumulants, Martingales

25 Pages Posted: 30 Jun 2020

See all articles by Peter Friz

Peter Friz

Technische Universität Berlin (TU Berlin)

Jim Gatheral

CUNY Baruch College

Rados Radoicic

CUNY Baruch College

Date Written: June 4, 2020

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 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

Friz, Peter and Gatheral, Jim and Radoicic, Rados, Forests, Cumulants, Martingales (June 4, 2020). Available at SSRN: https://ssrn.com/abstract=3620174 or http://dx.doi.org/10.2139/ssrn.3620174

Peter Friz (Contact Author)

Technische Universität Berlin (TU Berlin) ( email )

Straße des 17
Juni 135
Berlin, 10623
Germany

Jim Gatheral

CUNY Baruch College ( email )

Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States

Rados Radoicic

CUNY Baruch College ( email )

One Bernard Baruch Way
New York, NY 10010
United States

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