Lévy processes

A Lévy process is a stochastic process with stationary, independent increments starting at zero: the law of $X_t - X_s$ depends only on $t - s$ , and increments over disjoint intervals are independent. It is the canonical model for random motion that combines smooth drift, diffusion, and sudden jumps — the single framework that unifies Brownian motion, Poisson processes, and everything in between.

When it is relevant. Whenever a quantity accumulates by many small, independent contributions over time — a stock price with both continuous noise and discrete crashes, the total claims in an insurance portfolio, the net displacement of a particle in a turbulent medium — a Lévy process is the natural model. Its power is that one object, the Lévy–Khintchine triplet, encodes the entire law.

A Lévy process is a continuous-time Markov process, and its compensated version is a martingale. It builds on Random variables and expectation and Limit theorems.

Authoritative sources

K. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, 1999)

D. Applebaum, Lévy Processes and Stochastic Calculus (Cambridge University Press, 2009)

Lévy process — Wikipedia

Definition (cheatsheet)

$X_0 = 0$ , increments are stationary (law of $X_t - X_s$ depends only on $t-s$ ) and independent over disjoint intervals, and paths are continuous in probability (stochastic continuity).
Every Lévy process decomposes (Lévy–Itô decomposition) into drift + Brownian part + jump part: $X_t = b\,t + \sigma W_t + J_t$ , where $J_t$ is a pure-jump process built from a Poisson random measure.
The law is fully encoded in the characteristic exponent (Lévy–Khintchine triplet $(b, \sigma^2, \nu)$ ): $\mathbb{E}[e^{i\theta X_t}] = e^{t\,\psi(\theta)}$ with $\psi(\theta) = i b \theta - \tfrac{1}{2}\sigma^2\theta^2 + \int_{\mathbb{R}\setminus\{0\}} (e^{i\theta x} - 1 - i\theta x\, h(x))\,\nu(dx)$ , where $h(x) = x\,\mathbf{1}_{|x|<1}$ is a truncation function.
The Lévy measure $\nu$ describes jump intensities: $\nu(A)$ is the expected number of jumps per unit time with sizes in $A$ .

A common source of confusion. The drift parameter $b$ is not an absolute number: it depends on the truncation function $h$ you choose. Changing $h$ (for instance to $h(x) = x\,\mathbf{1}_{|x|\le 1}$ or $h(x) = x/(1+x^2)$ ) shifts $b$ by a compensating constant so that the triplet always describes the same process. Only the combination of $b$ and $\nu$ is invariant, not $b$ alone.

The Lévy–Khintchine triplet

Definition. A Lévy process is completely determined by its triplet $(b, \sigma^2, \nu)$ : a real drift $b$ , a diffusion coefficient $\sigma^2 \ge 0$ , and a Lévy measure $\nu$ (a measure on $\mathbb{R}\setminus\{0\}$ with $\int (1 \wedge x^2)\,\nu(dx) < \infty$ ). The characteristic exponent is

$\psi(\theta) = i b \theta - \tfrac{1}{2}\sigma^2\theta^2 + \int_{\mathbb{R}\setminus\{0\}} (e^{i\theta x} - 1 - i\theta x\, h(x))\,\nu(dx)$ .

Details

The whole theory hangs on one fact: stationary independent increments.

One increment is the whole process. Stationary independent increments mean you only ever need to know the law of one increment, say $X_1$ ; the rest follows by convolution. The characteristic function turns convolution into multiplication, so $\mathbb{E}[e^{i\theta X_t}] = e^{t\psi(\theta)}$ : the whole process is one function $\psi$ .
The Lévy–Khintchine theorem says every such $\psi$ has exactly the triplet form above: a linear drift term $b$ , a Gaussian term $\sigma^2$ , and an integral against the Lévy measure $\nu$ for jumps. Conversely, any choice of $(b, \sigma^2, \nu)$ with $\int (1 \wedge x^2)\,\nu(dx) < \infty$ produces a valid Lévy process.
Finite vs infinite activity. If $\nu(\mathbb{R}) < \infty$ , there are finitely many jumps per unit time (a compound Poisson part). If $\nu(\mathbb{R}) = \infty$ but $\int_{|x|<1} |x|\,\nu(dx) < \infty$ , jumps occur infinitely often but remain summable — the process still has finite variation. Dropping the last condition gives infinite-variation processes, where the “jumps” are so numerous and small that they blur into a continuous-looking drift.
Examples. The Poisson process has $\nu = \delta_1$ (unit jumps at rate 1); Brownian motion has $\nu = 0$ ; the Cauchy process has a heavy-tailed $\nu$ with infinite activity and infinite variation.

The truncation function $h(x) = x\,\mathbf{1}_{|x|<1}$ is what makes the integral converge. For large $x$ the term $e^{i\theta x} - 1$ is harmless, but near $x = 0$ the expansion $e^{i\theta x} - 1 \approx i\theta x$ would make $\int i\theta x\,\nu(dx)$ diverge when there are infinitely many small jumps. Subtracting $i\theta x\, h(x)$ cancels that divergence, and the leftover is absorbed into the drift $b$ — which is precisely why $b$ depends on $h$ .

Example problem

A compound Poisson process has jump rate $\lambda = 3$ per unit time and jump sizes $Y$ uniformly distributed on $\{1, 2\}$ (so $P(Y=1) = P(Y=2) = 1/2$ ). Write down its Lévy–Khintchine triplet and compute $E[X_t]$ and $\mathrm{Var}(X_t)$ .

Solution

The Lévy measure is $\nu = \lambda\,(\tfrac{1}{2}\delta_1 + \tfrac{1}{2}\delta_2) = \tfrac{3}{2}\delta_1 + \tfrac{3}{2}\delta_2$ , there is no diffusion ( $\sigma^2 = 0$ ), and with the standard truncation $h(x) = x\,\mathbf{1}_{|x|<1}$ (which vanishes on $\{1,2\}$ ) the drift is $b = 0$ . So the triplet is $(0, 0, \tfrac{3}{2}\delta_1 + \tfrac{3}{2}\delta_2)$ .

For a compound Poisson process, $E[X_t] = \lambda t\, E[Y]$ and $\mathrm{Var}(X_t) = \lambda t\, E[Y^2]$ . Here $E[Y] = \tfrac{3}{2}$ and $E[Y^2] = \tfrac{1}{2}(1) + \tfrac{1}{2}(4) = \tfrac{5}{2}$ , so

E[X_t] = 3t \cdot \tfrac{3}{2} = \tfrac{9}{2}\,t, \qquad \mathrm{Var}(X_t) = 3t \cdot \tfrac{5}{2} = \tfrac{15}{2}\,t.

Sample paths

Definition. The Lévy–Itô decomposition writes every Lévy process as $X_t = b\,t + \sigma W_t + J_t$ : a deterministic drift, a Brownian part $\sigma W_t$ , and a pure-jump part $J_t$ built from a Poisson random measure. The three terms are independent, and the path you see is their sum.

Details

Each ingredient has a very different look, and the decomposition is exactly what the four sample paths below show:

Drift $b\,t$ is a straight line — no randomness at all.
Brownian part $\sigma W_t$ is continuous but nowhere differentiable; it wanders with standard deviation $\sigma\sqrt{t}$ .
Jump part $J_t$ is a staircase: flat between jumps, with a vertical step at each jump. If $\nu(\mathbb{R}) < \infty$ the steps are isolated (compound Poisson); if $\nu(\mathbb{R}) = \infty$ the small jumps are so dense that they merge into a continuous-looking contribution.

The four panels below are one realization of each extreme of the family: pure diffusion, a unit-jump Poisson process, a compound Poisson process with random jump sizes, and pure drift.

The interactive simulation below lets you turn each ingredient on and off and watch the resulting path.

Example problem

A process has triplet $(b, \sigma^2, \nu) = (1, 4, 2\,\delta_{0.5})$ . Describe its path and compute $E[X_t]$ and $\mathrm{Var}(X_t)$ .

Solution

The path is a drift of slope $1$ , plus a Brownian motion with variance parameter $\sigma^2 = 4$ (so $\sigma = 2$ ), plus a compound Poisson process of rate $2$ with all jumps equal to $0.5$ .

For the drift, $E = t$ and $\mathrm{Var} = 0$ . For the Brownian part, $E = 0$ and $\mathrm{Var} = \sigma^2 t = 4t$ . For the compound Poisson part with jump size $c = 0.5$ and rate $\lambda = 2$ : $E = \lambda t\, c = 2t \cdot 0.5 = t$ , and $\mathrm{Var} = \lambda t\, c^2 = 2t \cdot 0.25 = 0.5\,t$ . The three parts are independent, so expectations add and variances add:

E[X_t] = t + 0 + t = 2t, \qquad \mathrm{Var}(X_t) = 0 + 4t + 0.5\,t = 4.5\,t.

Members of the family

Page	Drift	Diffusion	Jumps
Deterministic drift	yes	no	no
Brownian motion	optional	yes	no
Poisson process	optional	no	unit jumps
Compound Poisson process	optional	no	finite activity, arbitrary sizes
Variance-gamma process	optional	optional	infinite activity
CGMY process	optional	no	infinite activity, no diffusion
Stable process	optional	optional	infinite activity, self-similar

The first three rows are the building blocks; the last three are infinite-activity processes whose small jumps are too numerous to count individually. The stable processes are the self-similar members of the family, which ties into the Subordinators section below.

Subordinators

Definition. A subordinator is a non-decreasing Lévy process — a Lévy process with only positive jumps and no Brownian part, so its triplet is $(b, 0, \nu)$ with $\nu$ supported on $(0, \infty)$ . It is a “random clock”: a non-negative process $(T_t)$ that can be used to reparameterize (subordinate) another process.

Details

A subordinator is what you get when you strip a Lévy process down to its jump part and forbid negative jumps. Because it only moves upward, it can be read as a time change: given any process $(Y_s)$ , the composition $Y_{T_t}$ is called the subordination of $Y$ by $T$ .

The simplest subordinator is a Poisson process (unit positive jumps). A stable subordinator of index $\alpha \in (0,1)$ has Lévy measure $\nu(dx) = c\, x^{-1-\alpha}\,dx$ on $(0,\infty)$ and is self-similar: $T_{ct} \stackrel{d}{=} c^{1/\alpha} T_t$ . Subordinators are the bridge between the Lévy family and the self-similarity theme: a stable process is a Lévy process that is self-similar, and a stable subordinator is the time-change that produces it.

Example. Let $(N_t)$ be a Poisson process of rate $\lambda$ and $(W_s)$ a standard Brownian motion, independent. The subordinated process $W_{N_t}$ is a Lévy process: it is a compound Poisson process whose jumps are the Brownian increments between successive Poisson arrivals.

Example problem

Is $T_t = 2t + N_t$ (where $(N_t)$ is a rate-1 Poisson process) a subordinator? If so, write its triplet.

Solution

Yes. It is non-decreasing (the drift $2t$ and the counting process $N_t$ both only increase), it has stationary independent increments, and it starts at $0$ . Its triplet is $(b, \sigma^2, \nu) = (2, 0, \delta_1)$ : drift $b = 2$ , no diffusion, and unit jumps at rate $1$ .

Applications

Finance: jump-diffusion models (Merton, Kou) for asset prices with crashes.
Queues and insurance: compound Poisson processes model claim arrivals and aggregate claims (Cramér–Lundberg risk model).
Physics: anomalous diffusion, where a heavy-tailed $\nu$ produces superdiffusion.
Time changes: subordinators reparameterize other processes, linking the Lévy family to self-similar (stable) processes.

Brownian motion — the continuous, purely diffusive Lévy process.
Poisson processes — the pure-jump building block.
Martingales — compensated Lévy processes are martingales.
Parent topic: Stochastic processes.

Quizz

Try it yourself

A Lévy process has characteristic exponent

$\psi(\theta) = i\,\theta - \tfrac{1}{2}\,\theta^2 + \int_{\mathbb{R}\setminus\{0\}} (e^{i\theta x} - 1 - i\theta x\,\mathbf{1}_{|x|<1})\,\tfrac{1}{2}\delta_{3}(dx)$ .

Read off the triplet $(b, \sigma^2, \nu)$ .
Is the jump part finite- or infinite-activity?
Compute $E[X_1]$ and $\mathrm{Var}(X_1)$ .

Solution

1. Comparing with $\psi(\theta) = i b \theta - \tfrac{1}{2}\sigma^2\theta^2 + \int (e^{i\theta x} - 1 - i\theta x\, h(x))\,\nu(dx)$ with $h(x) = x\,\mathbf{1}_{|x|<1}$ : the drift is $b = 1$ , the diffusion is $\sigma^2 = 1$ , and the Lévy measure is $\nu = \tfrac{1}{2}\delta_3$ .

2. $\nu(\mathbb{R}) = \tfrac{1}{2} < \infty$ , so the jump part is finite-activity (a compound Poisson process of rate $\tfrac{1}{2}$ with jumps of size $3$ ).

3. The three parts are independent. Drift: $E = 1$ and $\mathrm{Var} = 0$ . Brownian ( $\sigma^2 = 1$ ): $E = 0$ and $\mathrm{Var} = 1$ . Compound Poisson (rate $\lambda = \tfrac{1}{2}$ , jump $c = 3$ ): $E = \lambda c = \tfrac{1}{2}\cdot 3 = \tfrac{3}{2}$ and $\mathrm{Var} = \lambda c^2 = \tfrac{1}{2}\cdot 9 = \tfrac{9}{2}$ . Adding:

E[X_1] = 1 + 0 + \tfrac{3}{2} = \tfrac{5}{2}, \qquad \mathrm{Var}(X_1) = 0 + 1 + \tfrac{9}{2} = \tfrac{11}{2}.

Lévy processes §

Definition (cheatsheet) §

The Lévy–Khintchine triplet §

Sample paths §

Members of the family §

Subordinators §

Applications §

Related pages §

Quizz §

Lévy processes

Definition (cheatsheet)

The Lévy–Khintchine triplet

Sample paths

Members of the family

Subordinators

Applications

Related pages

Quizz