Editing Lévy process (section)

== 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.