# Cumulants in Stochastic Analysis

## February 25 & 26, 2021

Cumulants have a long history. It was Thiele in 1889 who understood the role they play in probability and statistics. In this mini-workshop we discuss current developments around the notion of cumulants in stochastic analysis and non-commutative probability.

Sessions will be hosted via Zoom, with Gather.town breakout rooms. We suggest that you download the Gather.town app (Windows, macOS).

# Speakers & Discussants

Full list of participants.
- Michael ANSHELEVICH (Texas A&M University)
- Patric BONNIER (University of Oxford)
- Yvain BRUNED (Univeristy of Edinburgh)
- Ilya CHEVYREV (University of Edinburgh)
- Christa CUCHIERO (University of Vienna)
- Joscha DIEHL (University of Greifswald)
- Maxime FEVRIER (Paris-Saclay University)
- Uwe FRANZ (Université de Franche-Comté)
- Masaaki FUKASAWA (Osaka University)
- Jim GATHERAL (Baruch College)
- Malte GERHOLD (University of Greifswald)
- Nicolas GILLIERS (University of Greifswald)
- Paul HAGER (TU Berlin)
- Tom KLOSE (TU Berlin)
- Joachim KOCK (Autonomous University of Barcelona)
- Franz LEHNER (TU Graz)
- Alexandru NICA (University of Waterloo)
- Frédéric PATRAS (University of Nice Sophia-Antipolis)
- Nicolas PRIVAULT (Nanyang Technological University)
- Rados RADOIČIĆ (Baruch College)
- Josef TEICHMANN (ETH Zürich)

## Organizing Committee

- Kurusch EBRAHIMI-FARD (NTNU Trondheim)
- Peter K. FRIZ (WIAS/TU Berlin)
- Nikolas TAPIA (WIAS/TU Berlin)

# Schedule

All times CET (GMT+1)

# Abstracts

**Title:** Universality of affine and polynomial processes

**Abstract:** We show that generic classes of diffusion models are projections of infinite dimensional affine processes which coincide with polynomial processes in this setup. A key ingredient to establish this result is the signature process, well known from rough paths theory, which we use to prolong and linearize these classes of SDEs. The affine and polynomial technology then allows to get power series expansions in terms of the process' initial value both for the cumulants and the expected value of other analytic functions of the process' marginals. Our results apply in particular to so-called signature SDEs, a universal model class that has recently opened the door to more data-driven and more robust model selection mechanisms.

The talk is based on joint works with Sara Svaluto-Ferro and Josef Teichmann.

**Title:** Cumulant operators for continuous and discrete stochastic integrals

**Abstract:** We define cumulant operators for stochastic integrals using
the Malliavin calculus in a framework that includes Brownian and Poisson
stochastic integrals. Those operators are used to obtain moment and
cumulant expressions for stochastic integrals, with application to
Edgeworth-type expansions and probability approximation by the Stein
method. In the discrete setting of Poisson stochastic integrals,
applications include distribution estimation for shot noise processes,
random sets, and in the random-connection graph model.

**Title:** Realized cumulants for martingales

**Abstract:** Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and Friz et al. (2020).

**Title:** Signature cumulants and ordered partitions

**Abstract:** The sequence of so-called signature moments are known to characterize laws of many stochastic processes in analogy
with how the sequence of moments describes the laws of vector-valued random variables, which motivates us to study their
cumulant counterpart, so-called signature cumulants. To do so, we develop an elementary combinatorial approach and show
that in the same way that cumulants relate to the lattice of partitions, signature cumulants relate to the lattice of
so-called “ordered partitions”. We use this to give a new characterisation of independence of multivariate stochastic
processes and construct a family of unbiased minimum-variance estimators of signature cumulants.

**Title:** Cumulants for Lévy-type processes

**Abstract:** In a number of not-necessarily-commutative probability contexts, one can define the notion of cumulants appropriate to that theory, as shown by Lehner, Hasebe, and others. In this talk, I will look at a more restrictive setting, where we only consider joint distributions which are infinitely divisible within the context of the particular theory. We will discuss two approaches to cumulants of such distributions. First, the cumulant functional is the generator of the time-dependent family of moment functionals. Second, cumulants are expectations of higher variations of the process. Finally, the higher variations themselves share a number of properties of cumulants, even when the process does not have finite moments.

Many of the results in this talk are quite old; the recent ones are joint work with Zhichao Wang.

**Title:** Diamond trees and the forest expansion

**Abstract:** I will present a “broken exponential martingale” G-expansion that generalizes and unifies the earlier exponentiation result that I presented in Berlin in 2018. As one application, I show how to compute all terms in an expansion of the Lévy area. By reordering the trees in the G-expansion according to the number of leaves, our earlier exponentiation theorem can be recovered. As further applications, I will give model-free expressions for various quantities of interest under stochastic volatility. Finally, I will exhibit explicit computations of diamond trees under rough Heston.

#

**Title:** Cumulants, Spreadability and Quasisymmetric functions

**Abstract:** The notion of cumulants was extended to monotone convolution by Hasebe and Saigo. Contrary to the classical, free and Boolean case,
monotone independence does not satisfy the axiom of exchangeability
and thus mixed cumulants do not vanish.
It does satisfy the weaker axiom of spreadability
and in joint work with T.Hasebe we found that instead of vanishing, the mixed cumulants involve the coefficients of the Campbell-Baker-Hausdorff series.
More recently Novelli and Thibon pointed out an abstract interpretation of these cumulant identities in terms of Eulerian idempotents in the Hopf algebra WQSym of word quasi-symmetric functions.
We report on our joint effort to understand this correspondence.

**Title:** Shuffle algebras approach to operator-valued moments-cumulants relations.

**Abstract:** This talk will explain the shuffle algebras approach to operator-valued
moments-cumulants relations in non-commutative probability theory.
Shuffle algebras (Eilenberg & MacLane 1953) are ubiquitous and appear at
many places in mathematics, including perturbative quantum field theory,
rough path analysis, and more recently in non-commutative probability
theory.

I will start with a brief reminder on (free, boolean, monotone)
operator-valued probability theory. Then, I will show how nested
moments, used in the original definition of free cumulants by Speicher,
correspond to a representation of an operad on non-crossing partitions
introduced by Ebrahimi-fard, Foissy, Kock, Patras in 2020. Finally, by
using notions coming from $2$-monoidal category theory, I explain how
such representation solves a left half-shuffle fixed point equation.

**Title:** Moment cumulant formulae for multifaced universal products

**Abstract:** We discuss cumulants in the axiomatic approach to noncommutative independence via universal products. With each universal product, a class of partitions can be associated and the moment cumulant relation is encoded in certain exponential and logarithm maps strongly related to those partitions. We put a particular focus on the multifaced case (which includes, e.g., Voiculescu's bifreeness).

**Title:** On the use of Boolean cumulants in the study of free random variables

**Abstract:** A noncommutative probability space has a family of "free cumulant" functionals,
which are a powerful tool for studying operations with freely independent random
variables. On the same noncommutative probability space one can also consider
a family of "Boolean cumulant" functionals, defined with inspiration from the
parallel (but simpler) world of Boolean probability. Somewhat unexpectedly,
it turns out that Boolean cumulants can be competitive tools for certain free
probability calculations. I will present a joint work with M. Fevrier, M. Mastnak and
K. Szpojankowski (arXiv:1907.10842) where we show how Boolean cumulants can be
used in order to address the star-distribution of the product of two selfadjoint
freely independent random variables, and the distribution of the anti-commutator
of such random variables.

**Title:** Unified Signature Cumulants and Generalized Magnus Expansions

**Abstract:** The signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinite-dimensional, non-commutative ("Hausdorff") variation of Riccati's equation. Many examples are given. This is a joint work with Peter K. Friz and Nikolas Tapia.