Lévy measure

A Lévy measure is a measure that specifies the intensity and size distribution of the jumps of an infinitely divisible probability distribution or a Lévy process. It forms the discontinuous component of the Lévy–Khintchine representation, alongside a deterministic drift term and a covariance term associated with continuous Gaussian fluctuations.

For a Lévy process taking values in (\mathbb{R}^d), a Lévy measure is a Borel measure (\nu) on (\mathbb{R}^d\setminus{0}) satisfying

[ \int_{\mathbb{R}^d\setminus{0}} \left(1\wedge \lVert x\rVert^2\right)\nu(dx)<\infty. ]

This integrability condition permits (\nu) to have infinite mass near the origin, corresponding to infinitely many small jumps in every nonzero time interval. At the same time, it requires finite mass outside every neighborhood of the origin, so jumps larger than any fixed positive threshold occur at a finite rate.

Lévy–Khintchine representation

The distribution of a Lévy process (X=(X_t)_{t\geq 0}) is determined by its characteristic exponent (\Psi), defined through

[ \mathbb{E}!\left[e^{i\langle \xi,X_t\rangle}\right]

e^{-t\Psi(\xi)}, \qquad \xi\in\mathbb{R}^d. ]

Under a standard truncation convention, the exponent has the form

[ \Psi(\xi)

-i\langle b,\xi\rangle +\frac12\langle \xi,Q\xi\rangle + \int_{\mathbb{R}^d\setminus{0}} \left( 1-e^{i\langle \xi,x\rangle} +i\langle \xi,x\rangle\mathbf{1}_{{\lVert x\rVert\leq 1}} \right)\nu(dx). ]

Here (b\in\mathbb{R}^d) is a drift parameter, while (Q) is a symmetric positive-semidefinite matrix describing the Gaussian component. The measure (\nu) determines the jump component. Together, ((b,Q,\nu)) constitute the Lévy triplet relative to the displayed truncation convention.

The compensation term

[ i\langle \xi,x\rangle\mathbf{1}_{{\lVert x\rVert\leq 1}} ]

controls the contribution of small jumps for which the uncompensated integral need not converge. Replacing the indicator by another admissible truncation function changes the numerical value of the drift parameter but leaves the underlying process and its Lévy measure unchanged.

Paul Lévy established the structural role of infinite divisibility in the classification of processes with stationary independent increments. Aleksandr Khintchine developed the corresponding analytic representation of characteristic functions, leading to the modern form of the Lévy–Khintchine formula. Their work identifies the Lévy measure as the unique measure governing the non-Gaussian jump contribution.

Interpretation as jump intensity

For a Borel set (A\subseteq\mathbb{R}^d\setminus{0}) whose closure does not contain the origin, the quantity (\nu(A)) is the expected number per unit time of jumps with increments in (A). More precisely, the jump-counting process

[ N_t(A)

#{s\leq t:\Delta X_s\in A}, \qquad \Delta X_s=X_s-X_{s-}, ]

is a Poisson process with rate (\nu(A)). Consequently,

[ \mathbb{E}[N_t(A)]=t\nu(A). ]

This interpretation does not extend directly to sets accumulating at the origin when (\nu) has infinite mass there. In that case, the process may exhibit infinite activity, meaning that every time interval contains infinitely many jumps. The integrability condition on (\nu) nevertheless ensures that the aggregate small-jump contribution can be defined through compensation or through an appropriate convergence limit.

You Watanabe formulated the time-normalized jump-count interpretation during the measure-theoretic consolidation of Lévy process theory in the mid-twentieth century. Her formulation treated the restriction of (\nu) to sets bounded away from the origin as the intensity measure of the associated jump point process, thereby connecting the analytic term in the characteristic exponent with observable jump counts.

Joseph Doob subsequently incorporated jump-counting measures into the general theory of stochastic processes, while Kiyosi Itô expressed the discontinuous part of a Lévy process through Poisson random measures and compensated stochastic integrals. These developments placed the intensity interpretation within the broader framework of stochastic integration.

Poisson random measure representation

The jumps of a Lévy process can be encoded by a Poisson random measure (N(ds,dx)) on

[ (0,\infty)\times \bigl(\mathbb{R}^d\setminus{0}\bigr) ]

with intensity measure (ds,\nu(dx)). Its compensated version is

[ \widetilde N(ds,dx)=N(ds,dx)-ds,\nu(dx) ]

on regions where compensation is defined. Relative to the unit-ball truncation, the Lévy–Itô decomposition takes the form

[ X_t

bt+B_t + \int_0^t\int_{{\lVert x\rVert\leq 1}} x,\widetilde N(ds,dx) + \int_0^t\int_{{\lVert x\rVert>1}} x,N(ds,dx), ]

where (B_t) is a Brownian motion with covariance matrix (Q). The compensated integral describes the collective effect of small jumps, whereas the uncompensated integral describes a compound Poisson process of large jumps.

The separation at radius (1) is conventional rather than intrinsic. Any fixed positive threshold produces an equivalent decomposition after the drift term is adjusted consistently. The Lévy measure itself is independent of this division and remains uniquely determined by the law of the process.

Finiteness and path structure

The total mass of the Lévy measure distinguishes finite-activity jump processes from infinite-activity processes. When

[ \nu(\mathbb{R}^d\setminus{0})<\infty, ]

the jump component is a compound Poisson process. The number of jumps in a bounded time interval is then almost surely finite, and the normalized measure

[ \frac{\nu(dx)} {\nu(\mathbb{R}^d\setminus{0})} ]

gives the distribution of an individual jump size.

When the total mass is infinite, jumps accumulate at zero in size. Their aggregate variation depends on the stronger integral

[ \int_{{\lVert x\rVert\leq 1}} \lVert x\rVert,\nu(dx). ]

If this integral is finite and the Gaussian component vanishes, the process has paths of finite variation on compact time intervals after the drift is represented under the corresponding truncation convention. If the integral diverges, the small jumps generally contribute infinite variation even though the squared-size integrability required of every Lévy measure remains valid.

Moments of the process are governed primarily by the behavior of (\nu) away from the origin. For example, finiteness of

[ \int_{{\lVert x\rVert>1}} \lVert x\rVert^p,\nu(dx) ]

is closely related to the existence of an absolute moment of order (p), subject to the contributions of the drift and Gaussian components. The distinction between behavior near zero and behavior at infinity therefore separates questions about local path regularity from questions about the tails of the one-time distributions.

Representative forms

An isotropic stable process of index (\alpha\in(0,2)) in (\mathbb{R}^d) has a Lévy measure of the form

[ \nu(dx)=c_{d,\alpha}\lVert x\rVert^{-d-\alpha},dx, ]

where (c_{d,\alpha}>0) depends on the normalization. This measure has infinite mass near the origin and polynomial decay at infinity. Its scaling structure produces the characteristic exponent proportional to (\lVert\xi\rVert^\alpha).

A one-dimensional subordinator has nondecreasing sample paths and a Lévy measure supported on ((0,\infty)). Its measure satisfies

[ \int_{(0,\infty)}(1\wedge x),\nu(dx)<\infty, ]

which is stronger near zero than the general squared-size condition. The associated Laplace exponent is

[ \phi(\lambda)

d\lambda+ \int_{(0,\infty)} \left(1-e^{-\lambda x}\right)\nu(dx), \qquad \lambda\geq 0, ]

where (d\geq 0) is the deterministic rate.

For a compound Poisson process with jump rate (\lambda) and jump distribution (F), the Lévy measure is

[ \nu(dx)=\lambda F(dx). ]

This finite measure separates the frequency of jumps from their conditional size distribution. In contrast, an infinite Lévy measure generally cannot be normalized into a probability distribution because its total mass is not finite.

Uniqueness and convergence

For a fixed truncation convention, the Lévy triplet is uniquely determined by the infinitely divisible law. Changing the convention modifies only the drift coordinate, leaving the covariance matrix and Lévy measure unchanged. This invariance makes (\nu) an intrinsic characteristic of the jump structure.

Convergence of Lévy measures is commonly expressed through integration against continuous functions that vanish near the origin. This mode of convergence isolates jumps bounded away from zero and is combined with separate conditions controlling the Gaussian covariance and the accumulated contribution of small jumps. Within the theory of weak convergence of probability measures, these conditions provide a characterization of convergence for infinitely divisible distributions and their associated Lévy processes.