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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, a Lévy process, named after the French mathematician Paul Lévy, is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
with independent, stationary increments: it represents the motion of a point whose successive displacements are
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a
random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
. The most well known examples of Lévy processes are the
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
, often called the
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position ins ...
process, and the
Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have
discontinuous Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set o ...
paths. All Lévy processes are additive processes.


Mathematical definition

A
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
X=\ is said to be a Lévy process if it satisfies the following properties: # X_0=0 \,
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
; # Independence of increments: For any 0 \leq t_1 < t_2<\cdots , X_-X_, X_-X_,\dots,X_-X_ are mutually
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
; # Stationary increments: For any s, X_t-X_s \, is equal in distribution to X_; \, # Continuity in probability: For any \varepsilon>0 and t\ge 0 it holds that \lim_ P(, X_-X_t, >\varepsilon)=0. If X is a Lévy process then one may construct a
version Version may refer to: Computing * Software version, a set of numbers that identify a unique evolution of a computer program * VERSION (CONFIG.SYS directive), a configuration directive in FreeDOS Music * Cover version * Dub version * Remix * '' ...
of X such that t \mapsto X_t is
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
right-continuous with left limits.


Properties


Independent increments

A continuous-time stochastic process assigns a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''X''''t'' 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 ''X''''s'' − ''X''''t'' between its values at different times ''t'' < ''s''. To call the increments of a process independent means that increments ''X''''s'' − ''X''''t'' and ''X''''u'' − ''X''''v'' are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
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

To call the increments stationary means that the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
of any increment ''X''''t'' − ''X''''s'' 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 In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
, the probability distribution of ''X''''t'' − ''X''''s'' is normal with
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
0 and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
''t'' − ''s''. If X is a
Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
, the probability distribution of ''X''''t'' − ''X''''s'' is a
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
with expected value λ(''t'' − ''s''), where λ > 0 is the "intensity" or "rate" of the process.


Infinite divisibility

The distribution of a Lévy process has the property of
infinite divisibility Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matt ...
: given any integer ''n'', the
law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
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 assumptions 2 and 3. 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.


Moments

In any Lévy process with finite moments, the ''n''th moment \mu_n(t) = E(X_t^n), is a
polynomial function In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
of ''t''; these functions satisfy a binomial identity: :\mu_n(t+s)=\sum_^n \mu_k(t) \mu_(s).


Lévy–Khintchine representation

The distribution of a Lévy process is characterized by its
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at point ...
, which is given by the Lévy–Khintchine formula (general for all
infinitely divisible distribution In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characterist ...
s):
If X = (X_t)_ is a Lévy process, then its characteristic function \varphi_X(\theta) is given by :\varphi_X(\theta)(t) := \mathbb\left ^\right= \exp where a \in \mathbb, \sigma\ge 0, and \Pi is a -finite measure called the Lévy measure of X, satisfying the property :\int_ < \infty.
In the above, \mathbf is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
. Because characteristic functions uniquely determine their underlying probability distributions, each Lévy process is uniquely determined by the "Lévy–Khintchine triplet" (a,\sigma^2, \Pi). The terms of this triplet suggest that a Lévy process can be seen as having three independent components: a linear drift, a
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position ins ...
, and a Lévy jump process, as described below. This immediately gives that the only (nondeterministic) continuous Lévy process is a Brownian motion with drift; similarly, every Lévy process is a
semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the l ...
.


Lévy–Itô decomposition

Because the characteristic functions of independent random variables multiply, the Lévy–Khintchine theorem suggests that every Lévy process is the sum of Brownian motion with drift and another independent random variable, a Lévy jump process. The Lévy–Itô decomposition describes the latter as a (stochastic) sum of independent Poisson random variables. Let \nu=\frac— that is, the restriction of \Pi to \R\setminus(-1,1), renormalized to be a probability measure; similarly, let \mu=\Pi, _ (but do not rescale). Then :\int_=\Pi(\R\setminus(-1,1))\int_+\int_. The former is the characteristic function of a
compound Poisson process A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisson ...
with intensity \Pi(\R\setminus(-1,1)) and child distribution \nu. The latter is that of a compensated generalized Poisson process (CGPP): a process with countably many jump discontinuities on every interval a.s., but such that those discontinuities are of magnitude less than 1. If \int_<\infty, then the CGPP is a pure jump process. Therefore in terms of processes one may decompose X in the following way :X_t=\sigma B_t + at+Y_t+Z_t, t\geq 0, where Y is the compound Poisson process with jumps larger than 1 in absolute value and Z_t is the aforementioned compensated generalized Poisson process which is also a zero-mean martingale.


Generalization

A Lévy
random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other d ...
is a multi-dimensional generalization of Lévy process. Still more general are decomposable processes.


See also

*
Independent and identically distributed random variables In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
*
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
*
Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
* Gamma process *
Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
* Lévy flight


References

* * . * . * . {{DEFAULTSORT:Levy process Paul Lévy (mathematician)