Progressive Cumulants Framework in Stochastic Processes

• Progressive cumulants are a refined framework that sequentially constructs higher-order conditional cumulants using recursive combinatorial structures.
• They generalize classical cumulant analysis by incorporating filtration, time evolution, and martingale properties in probability and quantitative finance.
• This approach unifies disparate recursive formulas via set partitions and forest expansions, enabling practical estimation in stochastic and physical systems.

Progressive cumulants constitute an advanced framework for sequentially constructing and analyzing higher-order cumulants of random variables and stochastic processes, especially in settings with an underlying filtration or time evolution. This approach generalizes classical cumulant analysis to conditional and time-dependent contexts, capturing both formal and convergent expansions and unifying multiple previously disparate recursive formulae through combinatorial structures such as marked binary forests and Möbius set partitions. Progressive cumulant theory plays a fundamental role in probability theory, stochastic analysis, mathematical physics, and quantitative finance.

1. Progressive Cumulants in Filtrations

Let $(\Omega,\F,\{\F_t\},\P)$ be a filtered probability space with continuous martingales. Given an $\F_T$-measurable random variable $A_T$, define its time-$t$ conditional expectation $Y_t = \E[A_T|\F_t]$ and conditional cumulants $K^n_t(T)$ recursively by

$K^1_t(T) = \E_t[A_T], \qquad K^{n+1}_t(T) = \tfrac{1}{2} \sum_{k=1}^n (K^k \diamond K^{n+1-k})_t(T)$

where the diamond-product for semimartingales $A,B$ is

$(A \diamond B)_t(T) = \E \left[ \langle A, B \rangle_{T} - \langle A,B\rangle_{t} \mid \F_t \right]$

The cumulant expansion expresses

$\log \E [e^{z A_T} \mid \F_t] = \sum_{n=1}^N z^n K^n_t(T) + o(|z|^N), \quad z\to0$

for $A_T \in L^N$ and, under exponential integrability, converges absolutely for $|z| < \rho_t(\omega)$ (2002.01448). This recursion exactly recovers the conditional cumulants as semimartingale processes.

2. Broken Exponential Martingale Expansion and Forests

The "broken exponential martingale" expansion refines the cumulant framework. Setting

$Z_t(z_1, z_2) = z_1 Y_t + z_2 \langle Y \rangle_t$

one obtains

$\E[e^{Z_T(z)} | \F_t] = \exp\left\{ Z_t(z) + \Lambda^T_t(z) \right\}, \quad \Lambda^T_t(z) = \sum_{k\ge2} \mathcal{G}^k_t(T;z)$

with $\mathcal{G}^k$ recursively defined via powers-of-$z$ matching,

$$\forall k \geq 3\colon\quad \mathcal{G}^k_t(T;z) = \tfrac{1}{2} \sum_{j=2}^{k-2} \mathcal{G}^{k-j}_t(T;z) \mathcal{G}^j_t(T;z) + (z_1 Y \diamond \mathcal{G}^{k-1}(T;z))_t$$

Each term $\mathcal{G}^k$ is a sum over rooted binary forests, with combinatorial coefficients reflecting symmetries. Leaves are colored by types ($Y$ or $\langle Y \rangle$), tracking powers of $z_1$ and $z_2$ (2002.01448).

The classical cumulant recursion of Lacoin–Rhodes–Vargas arises as the one-dimensional slice $z_2=0$, yielding the time-$t$ conditional cumulants. The full two-parameter expansion encompasses both classical exponentiation results (Alòs–Gatheral–Radoičić) and cumulant recursion, related by forest reordering.

3. Systematic Progressive Construction via Set Partitions

For arbitrary random variables, the progressive (order-by-order) construction of cumulants leverages set partition combinatorics. The $n$th cumulant is given by the Möbius inversion formula: $$\langle Q_1 \ldots Q_n \rangle_c = \sum_{\pi\in\mathcal{P}(n)} \mu(\pi) \prod_{B\in\pi} \langle Q_B \rangle$$ where $\mathcal{P}(n)$ is the set of partitions of $\{1,\ldots,n\}$ and $\mu(\pi) = (-1)^{|\pi|-1} (|\pi|-1)!$. This enables systematic computation of cumulants for arbitrary observables and overlaps, with subtraction of self-correlations handled identically via set partitions (Francesco et al., 2016). Explicit pseudocode for cumulant computation is reducible to enumeration of set partitions and product of raw moments.

Order	Moment–Cumulant Formula	Comments
2	$\mu_2 - \mu_1^2$	Variance
3	$\mu_3 - 3\mu_2\mu_1 + 2\mu_1^3$	Skewness
4	$$\mu_4 - 4\mu_3\mu_1 - 3\mu_2^2 + 12\mu_2\mu_1^2 - 6\mu_1^4$$	Kurtosis

This recursive framework is computationally efficient for moderate $n$; the number of set partitions grows as the Bell number $B_n$ (Francesco et al., 2016). Progressive algorithms also facilitate automatic correction for detector non-uniformity and overlapping bins in multiparticle analyses.

4. Realized (Progressive) Cumulants for Martingales

For an $L^{n+1}$ martingale $M$, progressive (realized) cumulants generalize realized variance to arbitrary order utilizing complete Bell polynomials. For a partition $\pi$, the realized cumulant is

$$RC_{n+1}(M;\pi) = \sum_{i=1}^N B_{n+1}\left( \Delta M_{t_i}, \kappa_2(t_{i-1}), \ldots, \kappa_n(t_{i-1}), 0 \right)$$

where $\Delta M_{t_i} = M_{t_i} - M_{t_{i-1}}$ and $\kappa_j(t_{i-1})$ is the conditional cumulant at step $i-1$ (Fukasawa et al., 2020). These estimators have the aggregation property: sum over increments yields unbiased estimation of the marginal cumulant, independent of the partition.

Bell polynomial identities underpin both unbiasedness and recursive relations, e.g., for moments

$\E[(M_T - M_t)^k|\F_t] = B_k(\kappa_1(t,u), \kappa_2(t), ..., \kappa_k(t))$

Recursion for conditional cumulants is given by

$$d\,\kappa_{n+1}(t) = \sum_{j=1}^n \binom{n+1}{j+1} \kappa_{n+1-j}(t) d\kappa_j(t)$$

In finance, these realized cumulants estimate moment premia, facilitating comparison between physical and risk-neutral measures.

5. Edge-Dominated Progressive Cumulants in Coulomb Gases

For linear statistics in large-$N$ Coulomb gases with rotation-invariant potentials, higher-order progressive cumulants exhibit edge universality. Consider

${\cal L}_N = \sum_{i=1}^N f(|x_i|)$

Under large-$N$, the cumulant generating function is expressible via boundary data at the droplet edge ($r=R$), where $U'(R) R^{d-1} = 1$ (Bruyne et al., 2023). All cumulants beyond second order depend only on derivatives of $f$ and $U$ evaluated at $r=R$.

Specifically, for $q \ge 3$,

$$\kappa_q \simeq \beta^{q-1} N^{q-2} \left. \frac{d^{q-3}}{dr^{q-3}} \Bigg[ \frac{f'(r)}{ \beta (U''(r) + (d-1) U'(r)/r ) } r^{d-1} [f'(r)]^2 \Bigg] \right|_{r=R}$$

This universality applies in the regime of single-droplet support and smooth $f$; it breaks down if $f$ is non-smooth or density support is disconnected.

6. Applications and Worked Examples

The progressive cumulant framework underpins analysis in stochastic calculus (Lévy area, Itô integrals), mathematical physics, and finance. For example:

• Lévy Area: For planar Brownian motion $A_t = \int_0^t ( X_s dY_s - Y_s dX_s )$, the second progressive cumulant is

$K^2_t(T) = \tfrac{1}{2}(A \diamond A)_t(T) = \tfrac{1}{2} \int_t^T (\E_t[X_s^2] + \E_t[Y_s^2]) ds$

Higher cumulants follow recursions tied to the Taylor expansion of $\tan$, recovering Lévy's classic formulas (2002.01448).

• Iterated Integrals: For multi-words $a,b$, Itô covariation yields explicit formulas for diamond-products, enabling construction of forest expansions for arbitrary functionals.
• Forward-Variance Models (Finance): The broken-exponential expansion yields explicit triple moment generating functions for joint distribution of log-price, quadratic variation, and realized variance, collapsing to convolution-Riccati equations in affine models.

Worked numerical examples (see (Francesco et al., 2016)) demonstrate progressive subtraction of lower-order moments at each stage, maintaining combinatorial accuracy via Möbius weights.

7. Integrability and Domain of Validity

Progressive cumulant expansions admit both formal and convergent analytic regimes. Finite-moment expansion requires $A_T \in L^N$, yielding cumulants up to order $N$ (2002.01448). For full convergence, exponential-moment conditions suffice: $\exists\,\epsilon>0:\quad \E[ e^{x A_T} ] < \infty \quad \forall x \in (-\epsilon,\epsilon)$ Necessary and essentially sharp integrability criteria underlie the existence and convergence of cumulant and forest generating function expansions.

The progressive cumulant framework robustly unifies discrete, continuous, multivariate, and conditional cumulant computation, mapping classical cumulant analysis, forest expansions, and martingale realized statistics to a single recursive combinatorial structure.