Stochastic Processes

Levy-Ito Decomposition and Jump-Diffusion

Цели урока

State the three axioms of a Levy process and give examples
Apply the Levy-Khinchine theorem to characterize a process through the triplet (b, sigma^2, nu)
Interpret the Levy measure as jump intensity
Use the Levy-Ito decomposition for simulation and option pricing

Предварительные знания

Brownian motion
Poisson processes
Point processes

October 19, 1987. S&P 500 drops 22.6% in a single session. Black-Scholes probability of this: 10^{-160}. The number of atoms in the universe: 10^{80}. Brownian motion cannot model crashes. Levy processes can.

Bloomberg: exotic derivative pricing through Variance Gamma and CGMY
Risk management: VaR with heavy tails via alpha-stable distributions
Diffusion models: Levy noise for generating images with sharp textures
Telecommunications: self-similar traffic through alpha-stable Levy processes

Levy, Khinchine, and the mathematics of jumps

Paul Levy developed the theory of processes with independent increments in the 1930s-40s. Alexander Khinchine independently obtained the characteristic exponent in 1937. The joint theorem bears both names. The Levy-Ito decomposition was established in the 1950s. The financial breakthrough: Dilip Madan and colleagues introduced Variance Gamma and CGMY in the 1990s, displacing GBM for exotic derivatives.

Definition and Examples of Levy Processes

This lesson goes deeper into Levy processes: the focus is on the Levy-Ito decomposition, the Levy-Khintchine formula, and applications of jump-diffusion in option pricing and neural SDEs. October 19, 1987. The S&P 500 falls 22.6% in one day. Under the Black-Scholes model, the probability of this event is 10^{-160}. The number of atoms in the observable universe is 10^{80}. Brownian motion cannot describe catastrophes. Levy processes can.

The Variance Gamma process is used in the Madan-Carr-Chang (1998) option pricing model. Unlike Black-Scholes, VG captures the skewness and excess kurtosis of real return distributions.

A compound Poisson process is a special case of a Levy process: nu(dx) = lambda * F_jump(dx). Brownian motion is the limit of such processes as lambda tends to infinity with small jumps.

What distinguishes a Levy process from Brownian motion?

Levy processes are a broad class: Brownian motion, Poisson processes, compound Poisson processes. The key extension is the possibility of jumps while retaining independent stationary increments.

The Levy-Khinchine Theorem and the Ito Decomposition

Every Levy process consists of three components: drift, diffusion, and jumps. The Levy-Khinchine theorem states there is nothing else. Every infinitely divisible process with independent increments decomposes into these three parts - in exactly one way.

The CGMY financial model

A Levy process for exotic derivative pricing

The Carr-Geman-Madan-Yor (CGMY) process has Levy measure: nu(dx) = C * exp(-M*x) / x^{1+Y} for x > 0 and C * exp(-G*|x|) / |x|^{1+Y} for x < 0. Parameter Y in (0,2) controls small-jump activity. At Y approaching 2, the process approaches Brownian motion. Bloomberg uses CGMY for pricing exotic derivatives on equity indices.

Levy processes with infinite activity (nu(R \ {0}) = inf) have infinitely many jumps in any finite time interval. Sample paths are right-continuous with countably many discontinuities. Simulation requires discretizing the Levy measure.

What does the Levy measure nu describe in the triplet (b, sigma^2, nu)?

The Levy measure nu describes how often and how large jumps are. nu(A) = expected number of jumps landing in A per unit time. The integrability condition on nu guarantees the process exists.

Applications in Finance and Machine Learning

Stable Diffusion generates images by reversing a diffusion process. What if the noise is Levy rather than Gaussian? Alpha-stable Levy noise in diffusion models (2023) produces sharper textures and better captures rare details. Levy processes are not only finance.

Alpha-stable distributions (a special case of Levy without finite variance) are used in modern ML for noise injection: at alpha=2 this is Gaussian, at alpha<2 the tails are heavy. LeCun et al. (2023) use alpha-stable noise for regularization of neural networks.

Why does the exponential Levy model fit real market data better than GBM?

Real log returns have excess kurtosis > 3, skewness != 0, and rare large moves. GBM predicts a normal distribution. Levy models describe jumps and heavy tails through the Levy measure nu.

Connections to other topics

Levy processes unify diffusion, point processes, and financial mathematics

Financial mathematics — Related topic
Stochastic control — Related topic
SPDE — Related topic
ML: diffusion models — Related topic

Итоги

Levy process: X_0=0, independent stationary increments, continuity in probability
Levy-Khinchine: every Levy process is characterized by triplet (b, sigma^2, nu)
Levy measure nu(A) = expected number of jumps landing in A per unit time
Levy-Ito decomposition: drift + diffusion + large jumps + compensated small jumps

Вопросы для размышления

Why is Brownian motion a special case of a Levy process but not vice versa?
How does the Levy measure nu determine whether the process has finite or infinite activity?
Why are alpha-stable processes with alpha < 2 dangerous for risk management?

Связанные уроки

sp-23 — Point processes are the jump component of Levy processes
sp-20 — Brownian motion is the continuous part of a Levy process
sp-25-stochastic-control — Control of systems driven by Levy noise
sp-26-spde