Lévy process

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In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

The most well known examples of Lévy processes are Brownian motion and the Poisson process. Aside from Brownian motion with drift, all other proper Lévy processes have discontinuous paths.[citation needed]

Mathematical definition[edit]

A stochastic process X=\{X_t:t \geq 0\} is said to be a Lévy process if it satisfies the following properties:

  1. X_0=0 \, almost surely
  2. Independence of increments: For any 0 \leq t_1 < t_2<\cdots <t_n <\infty, X_{t_2}-X_{t_1}, X_{t_3}-X_{t_2},\dots,X_{t_n}-X_{t_{n-1}} are independent
  3. Stationary increments: For any s<t \,, X_t-X_s \, is equal in distribution to X_{t-s}. \,
  4. Continuity in probability: For any \epsilon>0 and t\ge 0 it holds that \lim_{h\rightarrow 0}P(|X_{t+h}-X_t|>\epsilon)=0

If X is a Lévy process then one may construct a version of X such that t \mapsto X_t is almost surely right continuous with left limits.


Independent increments[edit]

A continuous-time stochastic process assigns a random variable Xt to each point t ≥ 0 in time. In effect it is a random function of t. The increments of such a process are the differences XsXt between its values at different times t < s. To call the increments of a process independent means that increments XsXt and XuXv are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) independent.

Stationary increments[edit]

To call the increments stationary means that the probability distribution of any increment XtXs depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed.

If X is a Wiener process, the probability distribution of Xt − Xs is normal with expected value 0 and variance t − s.

If X is the Poisson process, the probability distribution of Xt − Xs is a Poisson distribution with expected value λ(t − s), where λ > 0 is the "intensity" or "rate" of the process.

Infinite divisibility[edit]

The distribution of a Lévy process has the property of infinite divisibility: given any integer "n", the law of a Lévy process at time t can be represented as the law of n independent random variables, which are precisely the increments of the Lévy process over time intervals of length t/n, which are independent and identically distributed by assumption.[clarification needed] Conversely, for each infinitely divisible probability distribution F, there is a Lévy process X such that the law of X_1 is given by F.


In any Lévy process with finite moments, the nth moment \mu_n(t) = E(X_t^n), is a polynomial function of t; these functions satisfy a binomial identity:

\mu_n(t+s)=\sum_{k=0}^n {n \choose k} \mu_k(t) \mu_{n-k}(s).

Lévy–Khintchine representation[edit]

The distribution of a Lévy process is characterized by its characteristic function, which is given by the Lévy–Khintchine formula: If  X = (X_t)_{t\geq 0} is a Lévy process, then its characteristic function  \phi_X(\theta) is given by

\phi_X(\theta) := \mathbb{E}\Big[e^{i\theta X_1} \Big] = \exp \Bigg( ai\theta - \frac{1}{2}\sigma^2\theta^2 + 
\int_{\mathbb{R}\backslash\{0\}} \big( e^{i\theta x}-1 -i\theta x \mathbf{I}_{|x|<1}\big)\,\Pi(dx) \Bigg)

where a \in \mathbb{R}, \sigma\ge 0, \mathbf{I} is the indicator function and \Pi is a sigma-finite measure called the Lévy measure of X, satisfying the property

\int_{\mathbb{R}\backslash\{0\}} 1 \wedge x^2  \Pi(dx) < \infty.

A Lévy process can be seen as having three independent components: a linear drift, a Brownian motion and a superposition of independent (centered) Poisson processes with different jump sizes; \Pi(dx) represents the rate of arrival (intensity) of the Poisson process with jump of size x. These three components, and thus the Lévy–Khintchine representation, are fully determined by the Lévy–Khintchine triplet (a,\sigma^2, \Pi). In particular, the only (nondeterministic) continuous Lévy process is a Brownian motion with drift.

Lévy–Itō decomposition[edit]

Any Lévy process may be decomposed into the sum of a Brownian motion, a linear drift and a pure jump process which captures all jumps of the original Lévy process. The latter can be thought of as a superposition of centered compound Poisson processes.This result is known as the Lévy–Itō decomposition.

Given a Lévy triplet (a,\sigma^2, \Pi) there exist three independent Lévy processes, which lie in the same probability space, X^{(1)}, X^{(2)}, X^{(3)} such that:

  • X^{(1)} is a Brownian motion with drift, corresponding to the absolutely continuous part of a measure and capturing the drift a and diffusion \sigma^2;
  • X^{(2)} is a compound Poisson process, corresponding to the pure point part of the singular measure W;
  • X^{(3)} is a square integrable pure jump martingale that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part of the singular measure W.

The process defined by X=X^{(1)}+X^{(2)}+X^{(3)} is then a Lévy process with triplet (a,\sigma^2, \Pi).

The process  X^{(3)} can be further decomposed as a sum of two independent processes the first pure jump zero mean martingale of jumps less than 1 In absolute value and the second a compound Poisson process describing the jumps bigger than one in absolute value.

See also[edit]