Editing Lévy process

{{short description|Stochastic process in probability theory}}
In [[probability theory]], a '''Lévy process''', named after the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]], is a [[stochastic process]]  with independent, stationary increments: it represents the motion of a point whose successive displacements are [[random variable|random]], 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]].

The most well known examples of Lévy processes are the  [[Wiener process]], often called the [[Brownian motion]] process, and the [[Poisson process]]. 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]] paths. All Lévy processes are [[additive process]]es.<ref>{{cite book |last1=Sato |first1=Ken-Iti |title=Lévy processes and infinitely divisible distributions |date=1999 |pages=31-68|publisher=Cambridge University Press |isbn=9780521553025}}</ref>

== Mathematical definition ==
A Lévy process is a [[stochastic process]] <math>X=\{X_t:t \geq 0\}</math> that satisfies the following properties:
# <math>X_0=0 \,</math> [[almost surely]];
# '''[[independent increments|Independence of increments]]:''' For any <math>0 \leq t_1 < t_2<\cdots <t_n <\infty</math>, <math>X_{t_2}-X_{t_1}, X_{t_3}-X_{t_2},\dots,X_{t_n}-X_{t_{n-1}}</math> are mutually [[Independence (probability theory)|independent]];
# '''[[Stationary increments]]:''' For any <math>s<t \,</math>, <math>X_t-X_s \,</math> is equal in distribution to <math>X_{t-s}; \,</math>
# '''[[Continuity in probability]]:''' For any <math>\varepsilon>0</math> and <math>t\ge 0</math> it holds that <math>\lim_{h\rightarrow 0} P(|X_{t+h}-X_t|>\varepsilon)=0.</math>

If <math>X</math> is a Lévy process then one may construct a [[version (probability theory)|version]] of <math>X</math> such that <math>t \mapsto X_t</math> is [[almost surely]] [[càdlàg|right-continuous with left limits]].

== Properties ==

=== Independent increments ===
A continuous-time stochastic process assigns a [[random variable]] ''X''<sub>''t''</sub> 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''<sub>''s''</sub> − ''X''<sub>''t''</sub> between its values at different times ''t'' < ''s''. To call the increments of a process '''independent''' means that increments ''X''<sub>''s''</sub> − ''X''<sub>''t''</sub> and ''X''<sub>''u''</sub> − ''X''<sub>''v''</sub> are [[Independence (probability theory)|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 independence|pairwise]]) independent.

=== Stationary increments ===
{{Main|Stationary increments}}
To call the increments stationary means that the [[probability distribution]] of any increment ''X''<sub>''t''</sub> − ''X''<sub>''s''</sub> depends only on the length ''t''&nbsp;−&nbsp;''s'' of the time interval; increments on equally long time intervals are identically distributed.

If <math>X</math> is a [[Wiener process]], the probability distribution of ''X''<sub>''t''</sub>&nbsp;−&nbsp;''X''<sub>''s''</sub> is [[normal distribution|normal]] with [[expected value]] 0 and [[variance]] ''t''&nbsp;−&nbsp;''s''.

If <math>X</math> is a [[Poisson process]], the probability distribution of ''X''<sub>''t''</sub>&nbsp;−&nbsp;''X''<sub>''s''</sub> is a [[Poisson distribution]] with expected value λ(''t''&nbsp;−&nbsp;''s''), where λ > 0 is the "intensity" or "rate" of the process.

If <math>X</math> is a [[Cauchy process]], the probability distribution of ''X''<sub>''t''</sub>&nbsp;−&nbsp;''X''<sub>''s''</sub> is a [[Cauchy distribution]] with density <math>f(x; t) = { 1 \over \pi } \left[ { \gamma \over x^2 + \gamma^2  } \right] </math> where <math>\gamma=t-s</math>.

=== Infinite divisibility ===
The distribution of a Lévy process has the property of [[Infinite divisibility (probability)|infinite divisibility]]: given any integer ''n'', the [[Law (stochastic processes)|law]] of a Lévy process at time t can be represented as the law of the sum 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 <math>F</math>, there is a Lévy process <math>X</math> such that the law of <math>X_1</math> is given by <math>F</math>.

=== Moments ===
In any Lévy process with finite [[moment (mathematics)|moments]], the ''n''th moment <math>\mu_n(t) = E(X_t^n)</math>, is a [[polynomial function]] of ''t''; [[binomial type|these functions satisfy a binomial identity]]:

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

== Lévy–Khintchine representation ==
The distribution of a Lévy process is characterized by its [[characteristic function (probability theory)|characteristic function]], which is given by the '''Lévy–Khintchine formula''' (general for all [[infinitely divisible distribution]]s):<ref>Zolotarev, Vladimir M. One-dimensional stable distributions. Vol. 65. American Mathematical Soc., 1986.</ref> <blockquote>If <math> X = (X_t)_{t\geq 0} </math> is a Lévy process, then its characteristic function <math> \varphi_X(\theta) </math> is given by
:<math>\varphi_X(\theta)(t) := \mathbb{E}\left[e^{i\theta X(t)}\right] = \exp{\left(t\left(ai\theta - \frac{1}{2}\sigma^2\theta^2 + \int_{\R\setminus\{0\}}{\left(e^{i\theta x}-1 -i\theta x\mathbf{1}_{|x|<1}\right)\,\Pi(dx)}\right)\right)}
</math>
where <math>a \in \mathbb{R}</math>, <math>\sigma\ge 0</math>, and  <math>\Pi</math> is a {{Mvar|&sigma;}}-finite measure called the '''Lévy measure''' of <math>X</math>, satisfying the property
:<math>\int_{\R\setminus\{0\}}{\min(1,x^2)\,\Pi(dx)} < \infty. </math>
</blockquote>

In the above, <math>\mathbf{1}</math> is the [[indicator function]].  Because [[Characteristic function (probability theory)|characteristic functions]] uniquely determine their underlying probability distributions, each Lévy process is uniquely determined by the "Lévy–Khintchine triplet" <math>(a,\sigma^2, \Pi)</math>.  The terms of this triplet suggest that a Lévy process can be seen as having three independent components: a linear drift, a [[Wiener process|Brownian motion]], 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]].<ref>Protter P.E. ''Stochastic Integration and Differential Equations.'' Springer, 2005.</ref>

=== 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 <math>\nu=\frac{\Pi|_{\R\setminus(-1,1)}}{\Pi(\R\setminus(-1,1))}</math>— that is, the restriction of <math>\Pi</math> to <math>\R\setminus(-1,1)</math>, normalized to be a probability measure; similarly, let <math>\mu=\Pi|_{(-1,1)\setminus\{0\}}</math> (but do not rescale).  Then 
:<math>\int_{\R\setminus\{0\}}{\left(e^{i\theta x}-1 -i\theta x\mathbf{1}_{|x|<1}\right)\,\Pi(dx)}=\Pi(\R\setminus(-1,1))\int_{\R}{(e^{i\theta x}-1)\,\nu(dx)}+\int_{\R}{(e^{i\theta x}-1-i\theta x)\,\mu(dx)}.</math>

The former is the characteristic function of a [[compound Poisson process]] with intensity <math>\Pi(\R\setminus(-1,1))</math> and child distribution <math>\nu</math>.  The latter is that of a [[compensated generalized Poisson process]] (CGPP): a process with countably many jump discontinuities on every interval [[Almost surely|a.s.]], but such that those discontinuities are of magnitude less than <math>1</math>.  If <math>\int_{\R}{|x|\,\mu(dx)}<\infty</math>, then the CGPP is a [[Pure jump model|pure jump process]].<ref>{{Citation|last=Kyprianou|first=Andreas E.|date=2014|pages=35–69|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-642-37632-0_2|isbn=9783642376313|title=Fluctuations of Lévy Processes with Applications|series=Universitext|chapter=The Lévy–Itô Decomposition and Path Structure}}</ref><ref>{{Cite web|url=http://www.math.uchicago.edu/~lawler/finbook2.pdf|title=Stochastic Calculus: An Introduction with Applications|last=Lawler|first=Gregory|author-link=Greg Lawler|date=2014|website=Department of Mathematics (The University of Chicago)|archive-url=https://web.archive.org/web/20180329130220/http://www.math.uchicago.edu/~lawler/finbook2.pdf|archive-date=29 March 2018|access-date=3 October 2018}}</ref> Therefore in terms of processes one may decompose <math>X</math> in the following way
:<math>X_t=\sigma B_t + at+Y_t+Z_t, t\geq 0,</math>
where <math>Y</math> is the compound Poisson process with jumps larger than <math>1</math> in absolute value and <math>Z_t</math> is the aforementioned compensated generalized Poisson process which is also a  zero-mean martingale.

== Generalization ==

A Lévy [[random field]] is a multi-dimensional generalization of Lévy process.<ref>{{Citation|last1=Wolpert|first1=Robert L. |last2=Ickstadt|first2=Katja |date=1998 |publisher=Springer, New York|language=en|doi=10.1007/978-1-4612-1732-9_12 |isbn=978-1-4612-1732-9 |title=Practical Nonparametric and Semiparametric Bayesian Statistics |series=Lecture Notes in Statistics|chapter=Simulation of Lévy Random Fields}}</ref><ref>{{Cite web|url=https://www2.stat.duke.edu/courses/Spring16/sta961/lec/levy.pdf |title=Lévy Random Fields |last=Wolpert|first=Robert L. 
 |date=2016|website=Department of Statistical Science (Duke University)}}</ref>
Still more general are decomposable processes.<ref>{{Cite journal | last = Feldman | first = Jacob | title = Decomposable processes and continuous products of probability spaces | journal = Journal of Functional Analysis | volume = 8 | issue = 1 | pages = 1–51 | date = 1971 | issn = 0022-1236 | doi = 10.1016/0022-1236(71)90017-6 | doi-access =  }}</ref>

== See also ==
* [[Independent and identically distributed random variables]]
*[[Wiener process]]
*[[Poisson process]]
*[[Gamma process]]
*[[Markov process]]
*[[Lévy flight]]

== References ==
{{Reflist}}
* {{Cite journal | last1 = Applebaum | first1 = David | title = Lévy Processes—From Probability to Finance and Quantum Groups | journal = Notices of the American Mathematical Society | volume = 51 | issue = 11 | pages = 1336–1347 | date = December 2004 | url = https://www.ams.org/notices/200411/fea-applebaum.pdf | issn = 1088-9477}}
* {{Cite book | last1 = Cont | first1 = Rama |last2 = Tankov | first2 = Peter| title = Financial Modeling with Jump Processes  | publisher = CRC Press | year = 2003 | isbn = 978-1584884132 }}.
* {{Cite book | last1 = Sato | first1 = Ken-Iti | title = Lévy Processes and Infinitely Divisible Distributions | publisher = Cambridge University Press | year = 2011 | isbn = 978-0521553025 }}.
* {{Cite book | last1 = Kyprianou | first1 = Andreas E. | title = Fluctuations of Lévy Processes with Applications. Introductory Lectures. Second edition. | publisher = Springer | year = 2014 | isbn = 978-3642376313 }}.

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