Infinitely Divisible Processes Overview

• Infinitely divisible processes are stochastic processes defined by finite-dimensional distributions that decompose into sums of i.i.d. components, characterized by Lévy–Khintchine representations.
• They encompass a broad class including Lévy, moving average, and selfsimilar processes, facilitating advanced analysis in path-space measures and semimartingale decompositions.
• Practical simulation techniques, such as shot-noise series and stochastic integrals, provide explicit error bounds and effective methods for modeling heavy-tailed and dependent structures.

An infinitely divisible process is a stochastic process such that for every integer $n\geq 1$, every finite-dimensional marginal admits a decomposition as the sum of $n$ independent identically distributed processes. This property connects the analysis of path-space measures, Lévy–Khintchine representations, series and integral decompositions, simulation techniques, and links to semimartingales, stable processes, and extremes of dependence.

1. Definition and General Structure

A random vector $X\in\mathbb{R}^d$ is infinitely divisible if, for each $n\in\mathbb{N}$, there exist i.i.d.\ random variables $X_{1,n},\dots,X_{n,n}$ such that $X\overset{d}=X_{1,n}+\cdots+X_{n,n}$. Every infinitely divisible law admits the Lévy–Khintchine representation: its characteristic function is

$$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$

with drift $a\in\mathbb{R}^d$, covariance $S\ge0$, and Lévy measure $\nu$ satisfying $n$0 [(Yuan et al., 2021), Section 1].

A stochastic process $n$1 is infinitely divisible if each finite-dimensional distribution is infinitely divisible. Lévy processes form a central subclass of infinitely divisible processes but the concept encompasses a much broader class including moving averages, selfsimilar processes, and time-changed or subordinated structures [(Yuan et al., 2021); (Hakassou et al., 2012); (Fonseca-Mora, 2017)].

2. Lévy–Khintchine and Lévy–Itô Representations

The infinite divisibility property extends the Lévy–Khintchine formula to path spaces and duals of nuclear spaces. For a process $n$2, the path-space law is determined by a triplet $n$3:

• $n$4: Gaussian covariance.
• $n$5: path-space Lévy measure on $n$6 satisfying integrability and regularity conditions.
• $n$7: drift function.

The finite-dimensional characteristic function is then

$n$8

[(Rosinski, 2016), Sec. 1].

In the dual of a nuclear space $n$9, one has the Lévy–Khintchine formula for infinitely divisible measures on $X\in\mathbb{R}^d$0 with drift $X\in\mathbb{R}^d$1, covariance form $X\in\mathbb{R}^d$2 on $X\in\mathbb{R}^d$3, and Lévy measure $X\in\mathbb{R}^d$4: $X\in\mathbb{R}^d$5 [(Fonseca-Mora, 2017), Sec. 3].

The Lévy–Itô decomposition gives

$X\in\mathbb{R}^d$6

where $X\in\mathbb{R}^d$7 is a Gaussian process, $X\in\mathbb{R}^d$8 is a Poisson random measure, and $X\in\mathbb{R}^d$9 its compensated version [(Fonseca-Mora, 2017), Sec. 4].

3. Series and Integral Representations

Shot-noise and Ferguson–Klass–Rosiński Series

Any infinitely divisible law with Lévy measure $n\in\mathbb{N}$0 (after suitable decomposition) admits a shot-noise series

$n\in\mathbb{N}$1

where $n\in\mathbb{N}$2 are Poisson points, $n\in\mathbb{N}$3 are i.i.d.\ marks, and $n\in\mathbb{N}$4 are centering shifts chosen for convergence. This representation is robust in multivariate and path-space settings, greatly facilitating simulation [(Yuan et al., 2021), Sec. 2].

Truncation after $n\in\mathbb{N}$5 terms yields a residual tail $n\in\mathbb{N}$6, whose mean-square (or higher moment) error can be explicitly evaluated: $n\in\mathbb{N}$7 where $n\in\mathbb{N}$8 is the truncated Lévy measure [(Yuan et al., 2021), Eq. (2)].

Stochastic Integrals and Moving Averages

General process representations are given as

$n\in\mathbb{N}$9

where $X_{1,n},\dots,X_{n,n}$0 is a Lévy process and $X_{1,n},\dots,X_{n,n}$1 is a suitable kernel. For stationary increment mixed moving averages (SIMMA),

$X_{1,n},\dots,X_{n,n}$2

where $X_{1,n},\dots,X_{n,n}$3 is an ID independently scattered random measure and $X_{1,n},\dots,X_{n,n}$4 is typically a deterministic kernel (Basse-O'Connor et al., 2012, Basse-O'Connor et al., 2014).

Time-changed and selfsimilar structures arise in additive processes with dilative stability: $X_{1,n},\dots,X_{n,n}$5 with $X_{1,n},\dots,X_{n,n}$6 itself constructed from an integral with respect to a background process and ultimately leading to a time-changed Lévy process representation [(Bhatti et al., 2016), Sec. 2].

4. Semimartingale and Path Regularity Criteria

A general result for infinitely divisible semimartingales $X_{1,n},\dots,X_{n,n}$7 representable as stochastic integrals against infinitely divisible random measures $X_{1,n},\dots,X_{n,n}$8: $X_{1,n},\dots,X_{n,n}$9 satisfies that $X\overset{d}=X_{1,n}+\cdots+X_{n,n}$0 is a semimartingale if and only if it admits a unique decomposition

$X\overset{d}=X_{1,n}+\cdots+X_{n,n}$1

where $X\overset{d}=X_{1,n}+\cdots+X_{n,n}$2 is càdlàg with independent increments (the “martingale” part) and $X\overset{d}=X_{1,n}+\cdots+X_{n,n}$3 is a predictable, finite variation process,

$X\overset{d}=X_{1,n}+\cdots+X_{n,n}$4

$X\overset{d}=X_{1,n}+\cdots+X_{n,n}$5

(Basse-O'Connor et al., 2012, Basse-O'Connor et al., 2014).

For stationary increment cases ($X\overset{d}=X_{1,n}+\cdots+X_{n,n}$6 a SIMMA), absolute continuity of the kernel in time (with suitable moment and integrability conditions for the jump and Gaussian parts) is necessary and sufficient for finite variation paths (Basse-O'Connor et al., 2012).

5. Sampling and Simulation Techniques

Shot-noise truncation provides efficient, practical sample-path generation:

• Fix truncation parameter $X\overset{d}=X_{1,n}+\cdots+X_{n,n}$7 (on Poisson intensity or absolute jump size).
• Simulate Poisson number of jumps and their marks.
• For SDEs driven by Lévy noise, a jump-adapted discretization combines exact simulation of jump times with strong-order schemes for drift and diffusion between jumps.

Error control is explicit: for $X\overset{d}=X_{1,n}+\cdots+X_{n,n}$8-stable processes, the number of jumps $X\overset{d}=X_{1,n}+\cdots+X_{n,n}$9 required for mean-square error $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$0 satisfies $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$1; for tempered or gamma processes, error decays exponentially allowing smaller $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$2 (Yuan et al., 2021, Kawai, 2021).

Small jumps (with controllable second moment) can be replaced by a Gaussian approximation, reducing computational cost [(Yuan et al., 2021), Sec. 6].

6. Infinitely Divisible Processes with Respect to Time and Extensions

A process $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$3 is infinitely divisible with respect to time (IDT) if for every $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$4,

$$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$5

where $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$6 are i.i.d. copies. Each one-dimensional marginal $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$7 is then classically infinitely divisible, and there exists a unique associated Lévy process with matching marginals [(Hakassou et al., 2012), Sec. 1].

Strong IDT processes admit the stronger property of path decomposition: $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$8 [(Mai et al., 2018), Sec. 1].

Multiparameter extensions (IDT of type 1 or 2) and weak IDT (equality in law only for fixed $$\varphi(\theta)=\mathbb{E}[e^{i\langle\theta,X\rangle}] =\exp\left( i\langle\theta,a\rangle - \tfrac12 \langle\theta,S\theta\rangle + \int_{\mathbb{R}_0^d} \left(e^{i\langle\theta,z\rangle} - 1 - i\langle\theta,z\rangle 1_{\{\|z\|\le 1\}}\right) \nu(dz) \right)$$9) extend the theory to a broader class, including classical Gaussian random fields and operator-stable fields (Hakassou et al., 2012).

7. Applications, Examples, and Extreme Models

Representative infinitely divisible processes:

• Fractional Lévy processes: moving averages with fractional kernels driven by Lévy processes, semimartingale if and only if suitable kernel regularity and small-jump moments are satisfied [(Basse-O'Connor et al., 2012); (Basse-O'Connor et al., 2012)].
• Stable and tempered stable Lévy motions: synthesized via LePage or shot-noise series, with simulation and error bounds explicit for heavy-tailed increments (Yuan et al., 2021, Kawai, 2021).
• Non-decreasing (subordinator) ID processes: Min- and max-infinite divisibility, exchangeable sequences with extreme-value and copula structure, constructed using stochastic process analogues of Marshall–Olkin or Archimedean models (Brück et al., 2020, Mai et al., 2018).
• Long-memory stationary infinitely divisible sequences: Ergodic-theoretic representations for heavy-tailed processes, with functional large deviation principles governed by null-recurrence and conservative flows [(Ghosh, 2010); (Owada, 2013)].

Extremal dependence structures, copulas, and the de Finetti-type correspondences further extend the reach of infinitely divisible processes in probabilistic dependence modeling (Brück et al., 2020).

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The theory of infinitely divisible processes underpins a wide spectrum of stochastic modeling, simulation, and limit theory. Their path properties—finite variation, semimartingale structure, and regularity—can be characterized through their Lévy–Khintchine triplets and explicit series/integral representations, with robust tools for simulation and broad applicability from stochastic integration to extreme-value analysis and multivariate dependence [(Yuan et al., 2021); (Rosinski, 2016); (Fonseca-Mora, 2017); (Kawai, 2021); (Hakassou et al., 2012); (Brück et al., 2020); (Mai et al., 2018)].