Lévy process
From Wikipedia, the free encyclopedia
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below. The most well-known examples are the Wiener process and the Poisson process.
Contents |
[edit] Properties
[edit] Independent increments
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 Xs − Xt between its values at different times t < s. To call the increments of a process independent means that increments Xs − Xt and Xu − Xv 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.
[edit] Stationary increments
To call the increments stationary means that the probability distribution of any increment Xs − Xt depends only on the length s − t of the time interval; increments with equally long time intervals are identically distributed.
In the Wiener process, the probability distribution of Xs − Xt is normal with expected value 0 and variance s − t.
In the Poisson process, the probability distribution of Xs − Xt is a Poisson distribution with expected value λ(s − t), where λ > 0 is the "intensity" or "rate" of the process.
[edit] Divisibility
The probability distributions of the increments of any Lévy process are infinitely divisible. There is a Lévy process for each infinitely divisible probability distribution.
[edit] Moments
In any Lévy process with finite moments, the nth moment 
is a polynomial function of t; these functions satisfy a binomial identity:
[edit] Lévy-Khinchin representation
It is possible to characterise all Lévy processes by looking at their characteristic function. This leads to the Lévy-Khinchin representation. If Xt is a Lévy process, then its characteristic function satisfies the following relation:
![\mathbb{E}\Big[e^{i\theta X_t} \Big] = \exp \Bigg( ait\theta - \frac{1}{2}\sigma^2t\theta^2 + t
\int_{\mathbb{R}\backslash\{0\}} \big( e^{i\theta x}-1 -i\theta x \mathbf{I}_{|x|<1}\big)\,W(dx) \Bigg)](../../../../math/0/c/7/0c78e0d34154fab238fab01966e50fbe.png)
where 
, 
and 
is the indicator function. The Lévy measure W must be such that
A Lévy process can be seen as comprising of three components: a drift, a diffusion component and a jump component. These three components, and thus the Lévy-Khinchin representation of the process, are fully determined by the Lévy-Khinchin triplet (a,σ2,W). So one can see that a purely continuous Lévy process is a Brownian motion with drift.
[edit] External links
- Applebaum, David (December 2004), “Lévy Processes—From Probability to Finance and Quantum Groups”, Notices of the American Mathematical Society (Providence, RI: American Mathematical Society) 51 (11): 1336-1347, ISSN 1088-9477, <http://www.ams.org/notices/200411/fea-applebaum.pdf>
