Levy Processes

A Levy process has stationary independent increments and continuous-in-probability paths, unifying Brownian motion, the Poisson process, and their combinations into a single family with jumps.

Definition

A Levy process is a stochastic process starting at zero with stationary and independent increments that is continuous in probability, so its increment over any interval has an infinitely divisible distribution and its characteristic exponent is given by the Levy-Khintchine formula.

Scope

This topic covers the definition of Levy processes through stationary independent increments, their correspondence with infinitely divisible distributions, the Levy-Khintchine formula decomposing the process into drift, Gaussian, and jump parts, the Levy-Ito decomposition of sample paths, subordinators and stable processes, and stochastic calculus and applications for processes with jumps.

Core questions

• What defines a Levy process and links it to infinitely divisible distributions?
• How does the Levy-Khintchine formula encode drift, diffusion, and jumps?
• How does the Levy-Ito decomposition describe the sample paths?
• What special Levy processes such as subordinators and stable processes arise?

Key theories

Levy-Khintchine formula

The characteristic function of a Levy process at any time is the exponential of a characteristic exponent comprising a linear drift, a Gaussian variance, and an integral against a Levy measure governing the jumps, giving a complete description of the law.

Levy-Ito decomposition

Every Levy process splits into a deterministic drift, an independent Brownian motion, and an independent pure-jump part built from a Poisson random measure of jumps, separating the continuous and discontinuous components of its paths.

Clinical relevance

Levy processes model asset returns with sudden jumps, insurance risk reserves, anomalous diffusion in physics, and queueing input with bursts, providing more realistic alternatives to purely Gaussian models wherever rare large movements matter.

History

De Finetti introduced infinitely divisible distributions in the 1920s, Levy and Khinchin derived the characteristic-exponent representation around 1934, and Ito's decomposition of the paths into continuous and jump parts completed the structural theory that bears their names, with renewed interest from mathematical finance since the 1990s.

Key figures

• Paul Levy
• Aleksandr Khinchin
• Kiyosi Ito
• Bruno de Finetti

Seminal works

• bertoin1996
• sato1999

Frequently asked questions

What unifies Brownian motion and the Poisson process?

Both are Levy processes, having stationary independent increments; Brownian motion is the continuous Gaussian case and the Poisson process is a pure-jump case, and general Levy processes combine drift, diffusion, and jumps.

What is the Levy measure?

It is the measure in the Levy-Khintchine formula that specifies the rate and sizes of the jumps of the process, controlling how often jumps of each magnitude occur.