Lévy Processes

Motivation: why this matters in quant finance

A Brownian motion model is powerful because it has independent and stationary increments. Those two assumptions make returns over non-overlapping periods easy to combine, and they are the reason characteristic functions, generators, and option-pricing equations stay tractable.

Continuity is a separate assumption. Lawler begins the jump-process chapter by asking what happens if we keep independent and stationary increments but stop requiring continuous paths. The resulting class is the Lévy processes. Brownian motion is still inside the class, but it is no longer the whole story.

For quantitative finance, this is the clean conceptual step from Black-Scholes toward jump models. A Lévy process is not "a fancy jump distribution"; it is the process-level structure that lets Brownian motion, Poisson processes, and compound Poisson shocks be handled with one set of tools.

The informal idea

A Lévy process has increments that are:

Independent: what happens after time $s$ is independent of the past once you look at the increment.
Stationary: the distribution of an increment depends on its length, not on calendar time.

Brownian motion adds continuous paths. Lévy processes do not require that. If continuity is dropped, the process may jump, while retaining the same clean time-additive structure that makes Brownian motion mathematically useful.

Lawler's first examples make this contrast concrete:

Brownian motion is the continuous-path Lévy process.
The Poisson process is the basic integer-valued jump Lévy process.
A compound Poisson process allows random jump sizes.
Hitting-time and Cauchy-process examples show that not every Lévy process looks Gaussian or Poisson.

Formal definitions

A stochastic process

X_t

is a Lévy process if it has independent and stationary increments: for

s,t>0

X_{s+t}-X_s

is independent of $\{X_r:r\le s\}$ and has the same distribution as $X_t-X_0$ .

Lawler also emphasises the path convention needed for a precise continuous-time object. The process is taken to be right-continuous with left limits, often called cadlag:

X_t=X_{t+}:=\lim_{q\downarrow t}X_q, \qquad X_{t-}:=\lim_{q\uparrow t}X_q.

At a jump time, $X_t\ne X_{t-}$ and the jump size is

\Delta X_t=X_t-X_{t-}.

Infinitely divisible increments

If $X_t$ is a Lévy process, then $X_1$ can be decomposed into $n$ i.i.d. increments:

X_1=\sum_{j=1}^n \left(X_{j/n}-X_{(j-1)/n}\right).

X_1

has an infinitely divisible distribution. Conversely, Lawler notes that every infinitely divisible distribution can arise as the time-one law of a Lévy process.

Two core examples are:

If $X\sim N(m,\sigma^2)$ , then $X$ is the sum of $n$ independent $N(m/n,\sigma^2/n)$ variables.
If $X$ is Poisson with mean $\lambda$ , then $X$ is the sum of $n$ independent Poisson variables with mean $\lambda/n$ .

This explains why normal and Poisson laws are the first two pillars of the Lévy family.

Characteristic exponents

Lawler uses characteristic exponents to express the additivity of independent increments. For a random variable $X$ ,

\mathbb{E}[e^{isX}]=e^{\Psi_X(s)}, \qquad \Psi_X(0)=0.

If $X$ and $Y$ are independent, then

\Psi_{X+Y}(s)=\Psi_X(s)+\Psi_Y(s).

For a Lévy process starting at the origin, stationary independent increments imply

\Psi_{X_t}(s)=t\Psi_{X_1}(s).

This is the main tractability property. The distribution over time $t$ is encoded by multiplying the one-period exponent by $t$ .

Poisson and compound Poisson examples

For a Poisson random variable with mean $\lambda$ ,

\Psi(s)=\lambda(e^{is}-1).

For a compound Poisson process with jump-size distribution $\mu^\#$ and rate $\lambda$ , define the finite measure

\mu=\lambda\mu^\#.

Then the characteristic exponent is

\Psi(s)=\int_{\mathbb{R}}(e^{isy}-1)\,d\mu(y).

This formula is the grounded version of the finance slogan "jumps add a tail term". The jump-size distribution enters through the integral, while the jump frequency is the total mass of $\mu$ .

Generators

A Lévy process is Markov, so it has a generator

Lf(x)=\lim_{t\downarrow 0}\frac{\mathbb{E}[f(X_t)\mid X_0=x]-f(x)}{t}.

For Brownian motion with drift $m$ and variance parameter $\sigma^2$ ,

Lf(x)=mf'(x)+\frac{1}{2}\sigma^2f''(x).

For a Poisson process with rate $\lambda$ ,

Lf(x)=\lambda[f(x+1)-f(x)].

For a compound Poisson process with Lévy measure $\mu$ ,

Lf(x)=\int_{\mathbb{R}}[f(x+y)-f(x)]\,d\mu(y).

If two independent Lévy processes are added, their generators add. For a Brownian motion with drift plus an independent unit-jump Poisson process,

Lf(x)=mf'(x)+\frac{1}{2}\sigma^2f''(x)+\lambda[f(x+1)-f(x)].

Worked examples

Example 1: Brownian plus Poisson exponent

Let

X_t=mt+\sigma W_t+N_t,

where $N_t$ is independent Poisson with rate $\lambda$ . The characteristic exponent of $X_1$ is

\Psi(s)=ims-\frac{1}{2}\sigma^2s^2+\lambda(e^{is}-1).

The first two terms come from the Gaussian component; the last term comes from unit jumps.

Example 2: compound Poisson mean and martingale compensation

Let $J_t=\sum_{i=1}^{N_t}Y_i$ with finite first moment and measure $\mu=\lambda\mu^\#$ . Then

m_J=\int_{\mathbb{R}}y\,d\mu(y)=\lambda\mathbb{E}[Y_1],

and

M_t=J_t-tm_J

is a martingale. This compensation is central when jump models are used for P&L: it separates predictable jump drift from unexpected jump variation.

Example 3: jump quadratic variation

For a compound Poisson process,

\langle J\rangle_t=\sum_{s\le t}(\Delta J_s)^2.

So a path with two jumps of sizes $-0.03$ and $0.05$ has jump quadratic variation

0.03^2+0.05^2=0.0034.

That is a realised path quantity, not a deterministic clock.

Common confusions and pitfalls

"Lévy process means jump process." No. Brownian motion is a Lévy process. Jumps enter when the continuous-path assumption is dropped.

"Stationary increments mean the process itself is stationary." No. The increment law depends only on the time length. The level $X_t$ usually changes distribution as $t$ grows.

"Every Lévy process has continuous paths except at rare jumps." Not from the definition alone. Lawler introduces cadlag paths so discontinuities can be handled, and later theory distinguishes finite-activity and more general jump behaviour.

"The Lévy measure in a compound Poisson process is a probability measure." In Lawler's compound Poisson setup, $\mu=\lambda\mu^\#$ is a finite measure whose total mass is the jump rate. The probability distribution is $\mu^\#$ .

"Characteristic exponents are decorative notation." They carry the main additivity. Independent components add exponents, and time scales them by $t$ .

Where this goes next

Poisson Processes: Develops the basic finite-activity jump clock.
Jump-Diffusion Processes: Combines Brownian and compound Poisson parts in finance-facing models.
Quadratic Variation: Explains why Brownian variation and jump variation behave differently.
Infinitesimal Generators and Kolmogorov Equations: Turns the generator formulas into evolution equations.
Local Martingales and Semimartingales: Places jump and diffusion components in the broader calculus of semimartingales.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 6 §6.1 (Lévy processes), §6.2 (Poisson process), §6.3 (Compound Poisson process).

title: Lévy Processes order: 7 verified: false latest_update_date: 26/04/2026 progress: completed difficulty: advanced prerequisites:

learn/math/probability/stochastic-processes/brownian-motion
learn/math/probability/stochastic-processes/jump-diffusion-processes
learn/math/probability/foundations/random-variables

Lévy Processes

Motivation: why this matters in quant finance

Lévy processes provide the standard framework for continuous-time models with independent and stationary increments. They include Brownian Motion, Jump-Diffusion Processes, Poisson processes, compound Poisson processes, and infinite-activity jump models under one language.

In quantitative finance, the payoff is practical: Lévy models can keep the tractable independent-increment structure of Brownian motion while allowing discontinuities, skew, and heavier tails. They form the mathematical backbone of exponential Lévy option models and clarify exactly what is added when Black-Scholes is extended with jumps.

Lawler's construction frames the contrast sharply: Brownian motion is the continuous-path member of the Lévy family, while Poisson and compound Poisson processes are the basic discontinuous members. The lesson's job is to understand that common structure before moving to generators, characteristic exponents, or jump-pricing equations.

Definition and Basic Properties

Formal Definition

A stochastic process

\{X_t\}_{t \geq 0}

is a Lévy process if:

Initial condition: $X_0 = 0$ almost surely
Independent increments: For $0 \leq t_1 < t_2 < \cdots < t_n$ , the random variables $X_{t_1}, X_{t_2} - X_{t_1}, \ldots, X_{t_n} - X_{t_{n-1}}$ are mutually independent
Stationary increments: $X_{t+h} - X_t$ has the same distribution as $X_h$ for all $t, h > 0$
Stochastic continuity: $\lim_{h \to 0} \mathbb{P}(|X_{t+h} - X_t| > \varepsilon) = 0$ for all $\varepsilon > 0$ and $t \geq 0$
Càdlàg paths: Sample paths are right-continuous with left limits

Examples

Brownian Motion: $X_t = \sigma W_t + \mu t$
Compound Poisson Process: $X_t = \sum_{i=1}^{N_t} Y_i$
Gamma Process: Increasing Lévy process with Gamma-distributed increments
Inverse Gaussian Process: Subordinator used in time-changed models
Variance Gamma Process: Time-changed Brownian motion

The Lévy-Khintchine Formula

Characteristic Function

The fundamental theorem characterizing Lévy processes states that the characteristic function of $X_t$ has the form:

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

where the Lévy exponent

\psi(u)

is given by:

\psi(u) = i\gamma u - \frac{1}{2}\sigma^2 u^2 + \int_{\mathbb{R} \setminus \{0\}} \left(e^{iuz} - 1 - iuz\mathbf{1}_{|z|<1}\right) \nu(dz)

Lévy Triplet

Every Lévy process is uniquely determined by its Lévy triplet

(\gamma, \sigma^2, \nu)

$\gamma \in \mathbb{R}$ : Drift coefficient
$\sigma^2 \geq 0$ : Gaussian (diffusion) coefficient
$\nu$ : Lévy measure on $\mathbb{R} \setminus \{0\}$ satisfying $\int_{\mathbb{R}} (1 \wedge |z|^2) \nu(dz) < \infty$

Lévy Measure

The Lévy measure

\nu

encodes the jump structure:

$\nu(\{0\}) = 0$ (no jumps of size zero)
$\nu(A)$ represents the expected number of jumps in set $A$ per unit time
$\int_{|z|>1} \nu(dz) < \infty$ ensures finite jump activity for large jumps
$\int_{|z| \leq 1} |z|^2 \nu(dz) < \infty$ ensures finite quadratic variation from small jumps

Lévy-Itô Decomposition

Decomposition Formula

Any Lévy process can be written as:

X_t = \gamma t + \sigma W_t + \lim_{\varepsilon \to 0} \left[\sum_{0 < s \leq t, |\Delta X_s| > \varepsilon} \Delta X_s + \sum_{0 < s \leq t, |\Delta X_s| \leq \varepsilon} (\Delta X_s - \mathbb{E}[\Delta X_s])\right]

More precisely:

X_t = \gamma t + \sigma W_t + X_t^{(1)} + X_t^{(2)}

where:

$\gamma t$ : Linear drift
$\sigma W_t$ : Brownian motion component
$X_t^{(1)}$ : Compensated sum of small jumps
$X_t^{(2)}$ : Sum of large jumps

Small and Large Jumps

Small jumps ( $|z| \leq 1$ ): Infinite activity, compensated to have zero mean
Large jumps ( $|z| > 1$ ): Finite activity, uncompensated

This decomposition shows that Lévy processes are the most general processes with independent increments.

Special Cases and Examples

1. Subordinators

A Lévy process with non-decreasing paths. Examples:

Gamma process: $\Gamma_t \sim \text{Gamma}(\alpha t, \beta)$
Inverse Gaussian process: Used in variance gamma models
Stable subordinator: Heavy-tailed increasing process

2. Symmetric α-Stable Processes

Characterized by stability under addition and heavy tails:

\psi(u) = -c|u|^\alpha, \quad 0 < \alpha \leq 2

When $\alpha = 2$ , this reduces to Brownian motion.

3. Variance Gamma Process

X_t = \theta G_t + \sigma W_{G_t}

where $G_t$ is a Gamma subordinator. This creates infinite activity pure-jump processes popular in option pricing.

4. Normal Inverse Gaussian (NIG)

Has Lévy density:

\nu(x) = \frac{\alpha\delta}{\pi|x|} K_1(\alpha|x|) e^{\beta x}

where $K_1$ is the modified Bessel function of the first kind.

Time Changes and Subordination

Definition

A subordinator is an increasing Lévy process

G_t

with

G_0 = 0

. Time-changing a process

Y_t

by a subordinator creates:

X_t = Y_{G_t}

Examples

Variance Gamma: $X_t = W_{G_t}$ where $G_t$ is Gamma
Normal Inverse Gaussian: Uses inverse Gaussian subordinator
CGMY: Uses a different subordination scheme

Financial Interpretation

Time changes model varying market activity:

Business time: Markets are more active during certain periods
Transaction time: Time measured in number of trades
Information time: Time measured by information arrival

Infinitely Divisible Distributions

Definition

A random variable

X

is infinitely divisible if for every

n \geq 1

, there exist i.i.d. random variables

X_1^{(n)}, \ldots, X_n^{(n)}

such that:

X \stackrel{d}{=} X_1^{(n)} + \cdots + X_n^{(n)}

Connection to Lévy processes

The distribution of $X_t$ for any Lévy process is infinitely divisible.
Every infinitely divisible distribution corresponds to a Lévy process.
Normal and Poisson laws are the canonical examples: Brownian motion decomposes normal increments, while Poisson processes decompose event counts.

This is the structural reason Lévy processes are not an arbitrary model class. They are exactly the continuous-time processes whose fixed-horizon increments can be split consistently into any number of smaller independent increments.

Option Pricing with Lévy Processes

Exponential Lévy Models

Asset prices follow:

S_t = S_0 e^{X_t}

where $X_t$ is a Lévy process with appropriate drift adjustment.

Risk-Neutral Pricing

Under the risk-neutral measure, the discounted price must be a martingale:

e^{-rt}S_t = e^{-rt}S_0 e^{X_t}

This requires the Lévy exponent to satisfy:

\psi(-i) = r

Fourier Methods

Option prices can be computed using characteristic functions:

C = e^{-rT} \int_0^\infty (S_T - K)^+ f_{S_T}(s) ds

where $f_{S_T}$ is recovered via inverse Fourier transform.

Popular Models

Black-Scholes: Geometric Brownian motion
Merton Jump-Diffusion: Brownian motion + compound Poisson
Variance Gamma: Pure jump model
CGMY: Infinite activity model
NIG: Solvable in terms of special functions

Simulation Methods

Direct Simulation

Simulate large jumps: Poisson process for $\nu(\{|z| > \varepsilon\})$
Simulate small jumps: Use series representation or approximation
Add Gaussian component: Standard Brownian motion
Add drift: Deterministic term

Series Representation

For infinite activity processes:

X_t = \sigma W_t + \sum_{i=1}^{\infty} z_i \mathbf{1}_{U_i \leq t}

where $U_i$ are arrival times and $z_i$ are jump sizes.

Acceptance-Rejection

For processes with known Lévy densities, use acceptance-rejection to simulate jump sizes.

Estimation from Data

Maximum Likelihood

For discretely observed Lévy processes, maximize:

L(\theta) = \prod_{i=1}^n f_{\Delta}(X_{t_i} - X_{t_{i-1}}; \theta)

where $f_\Delta$ is the transition density over interval $\Delta$ .

Method of Moments

Match sample moments to theoretical moments derived from the characteristic function.

Generalized Method of Moments (GMM)

Use multiple moment conditions based on the Lévy-Khintchine formula.

Applications in Risk Management

Value at Risk (VaR)

Lévy models better capture:

Heavy tails: More accurate extreme quantile estimation
Asymmetry: Different behavior for gains vs. losses
Jump risk: Sudden large moves

Expected Shortfall (ES)

\text{ES}_\alpha = \mathbb{E}[X | X > \text{VaR}_\alpha]

Lévy models provide more realistic tail risk measures.

Operational Risk

Compound Poisson processes model operational loss frequency and severity.

Advanced Topics

Fluctuation Theory

Studies the behavior of suprema, infima, and first passage times of Lévy processes.

Lévy Copulas

Extend the concept to multivariate settings with dependent marginal Lévy processes.

Path-Dependent Options

Barrier options, lookback options, and Asian options under Lévy dynamics.

Credit Risk Models

Jump-to-default models use Lévy processes to model credit events.

Common confusions and pitfalls

"Lévy process means jump process." No. Brownian motion is a Lévy process. Jumps enter only when the continuous-path assumption is dropped.

"Stationary increments mean the process is stationary." No. The increments have time-homogeneous laws; the level

X_t

usually changes distribution with

t

"Infinite activity means infinitely many large jumps." No. Infinite-activity models can have infinitely many small jumps on finite intervals while still having finite large-jump activity.

"A Lévy model is automatically risk-neutral." No. It is a probabilistic model class. Risk-neutrality requires a measure under which discounted prices are martingales.

"The Lévy measure is an ordinary probability distribution." Not generally. It measures jump intensity by size, can have infinite total mass near zero, and is constrained by integrability conditions rather than total probability one.

Where this goes next

Lévy processes unify many concepts in stochastic finance:

Generalize Brownian Motion and Jump-Diffusion
Foundation for Poisson processes and compound Poisson models
Enable realistic modeling beyond Geometric Brownian Motion
Connect to heavy-tailed distributions and extreme value theory
Basis for time-changed processes and subordination
Framework for incomplete market option pricing models

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 6 §6.1 (Lévy processes), §6.2 (Poisson process), §6.3 (Compound Poisson process).