Levy Processes — Mathematics Knowledge System

Overview

A Levy process is a stochastic process with stationary and independent increments that starts at zero and has cadlag paths (right-continuous with left limits). Levy processes generalise both Brownian motion (continuous paths) and the Poisson process (jump paths) and can be thought of as combinations of a linear drift, a Brownian motion component, and a compensated jump component. The Levy-Khintchine formula characterises all Levy processes via their characteristic exponent. They are the building blocks of jump-diffusion models in mathematical finance.

Intuition

A Levy process is like a random walk in continuous time where the steps can be either: (1) infinitesimally small and Gaussian (Brownian motion), (2) finitely many big jumps (compound Poisson), or (3) infinitely many small jumps (e.g., alpha-stable processes). The key property is that increments are independent: what happens over [s,t] tells you nothing about [t,u]. The characteristic function E[e^{iu*X_t}] = e^{t*psi(u)} has a fixed structure given by the Levy-Khintchine formula.

Formal Definition

Definition

A cadlag stochastic process X = (X_t)_{t>=0} with X_0 = 0 is a Levy process if it has independent increments (X_{t+s} - X_t independent of sigma(X_r, r<=t)) and stationary increments (X_{t+s} - X_t has the same distribution as X_s). The Levy-Khintchine formula: E[e^{iu*X_t}] = e^{t*psi(u)} where psi(u) = i*b*u - (sigma^2/2)*u^2 + integral_{R\{0}} (e^{iux} - 1 - iux*1_{|x|<1}) nu(dx) for Levy measure nu.

\mathbb{E}[e^{iu X_t}] = e^{t\psi(u)}

Characteristic function via characteristic exponent

\psi(u) = ibu - \frac{\sigma^2}{2}u^2 + \int_{\mathbb{R}\setminus\{0\}}\!\left(e^{iux} - 1 - iux\mathbf{1}_{|x|<1}\right)\nu(dx)

Levy-Khintchine formula

\int_{\mathbb{R}\setminus\{0\}} (|x|^2 \wedge 1)\, \nu(dx) < \infty

Levy measure condition

X_t = bt + \sigma W_t + \int_0^t\!\int_{|x|<1} x\, \tilde{N}(ds,dx) + \int_0^t\!\int_{|x|\ge 1} x\, N(ds,dx)

Levy-Ito decomposition

Notation

Notation	Meaning
$\psi(u)$	Characteristic exponent of Levy process
$\nu$	Levy measure (describes jump intensity)
$N(ds,dx)$	Poisson random measure on [0,inf) x R
$\tilde{N}(ds,dx) = N(ds,dx) - ds\,\nu(dx)$	Compensated Poisson random measure

Theorems

Theorem 1: Levy-Ito Decomposition

Every Levy process X can be uniquely written as X_t = b*t + sigma*W_t + X^{(1)}_t + X^{(2)}_t where W is a Brownian motion, X^{(1)} is a compound Poisson process of large jumps (|x| >= 1), and X^{(2)} is a square-integrable martingale of small jumps (|x| < 1). The four components are independent.

Theorem 2: Levy-Khintchine Theorem

A probability distribution mu is infinitely divisible (mu = mu_{1/n}^{*n} for some mu_{1/n} and all n) if and only if its characteristic function is of the form exp(psi(u)) where psi is given by the Levy-Khintchine formula with parameters (b, sigma^2, nu). The triplet (b, sigma^2, nu) uniquely determines the Levy process.

Theorem 3: Stable Processes and Self-Similarity

A Levy process X is alpha-stable (0 < alpha <= 2) if X_{ct} has the same distribution as c^{1/alpha} X_t for all c > 0. For alpha = 2 this is Brownian motion; for alpha < 2 the Levy measure is nu(dx) = C|x|^{-1-alpha} dx, giving heavy-tailed jumps with no finite variance for alpha < 2.

Worked Examples

1
The Poisson process N_t has independent and stationary increments and starts at 0 -- it is a Levy process.
2
Characteristic function: E[e^{iu*N_t}] = exp(t*lambda*(e^{iu} - 1)).
$\mathbb{E}[e^{iuN_t}] = e^{t\lambda(e^{iu}-1)}$
3
This matches the Levy-Khintchine formula with b = 0, sigma = 0, nu = lambda * delta_1 (point mass at 1).
$\psi(u) = \lambda(e^{iu} - 1 - iu) + iu\lambda \text{ (splitting compensation)}$
4
Levy measure: nu = lambda*delta_1 (one jump of size 1 at Poisson rate lambda). Triplet: (b=0, sigma=0, nu = lambda*delta_1) adjusted for the compensation convention.

✓ Answer

The Poisson process is a Levy process with Levy measure nu = lambda*delta_1, no Gaussian component, and drift b adjusted for the compensator.

Practice Problems

Mediumfree response

Explain the Levy-Ito decomposition and why Levy processes can be decomposed into a Brownian part and a jump part.

Common Mistakes

Common Mistake

Thinking Levy processes always have finite activity (finitely many jumps per unit time).

Many Levy processes have infinite activity: infinitely many small jumps per unit time (e.g., alpha-stable processes, variance gamma). The Levy measure can have infinite total mass near 0, corresponding to infinite activity.

Quiz

Which statement about Levy processes is correct?

All Levy processes have continuous sample pathsLevy processes have stationary and independent incrementsLevy processes are always martingalesLevy processes require a finite Levy measure

Historical Background

Paul Levy studied processes with independent and stationary increments in the 1930s-40s, building on work by Kolmogorov. The Levy-Khintchine formula (1934) gave a complete characterisation of infinitely divisible distributions and their associated Levy processes. Ito (1942) decomposed Levy processes into a Brownian part and a jump part (Ito's decomposition). The Levy-Ito decomposition is the fundamental structure theorem of Levy processes.

1934
Levy and Khintchine independently characterise infinitely divisible distributions
Paul Levy, Alexandr Khintchine
1942
Ito decomposes Levy processes into Gaussian and jump components
Kiyosi Ito
1950s
Bochner and Feller further develop the theory via semigroups
Salomon Bochner, William Feller

Summary

Levy processes have stationary independent increments; they generalise both Brownian motion and Poisson processes.
The Levy-Khintchine formula characterises all Levy processes via (b, sigma^2, nu): drift, Gaussian variance, and jump measure.
The Levy-Ito decomposition: X = drift + Brownian + big jumps (compound Poisson) + small jumps (martingale).
Alpha-stable processes are Levy processes with self-similar heavy-tailed distributions.

References

BookApplebaum, D. Levy Processes and Stochastic Calculus. Cambridge, 2009.
BookCont, R. and Tankov, P. Financial Modelling with Jump Processes. CRC Press, 2004.
WebsiteWikipedia -- Levy process