LéVY PROCESS

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.
A continuous-time stochastic process assigns a random variable ''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 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. To call the increments 'stationary' means that the probability distribution of any increment ''X''_''s'' − ''X''_''t'' 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 ''X''_''s'' − ''X''_''t'' is normal with expected value 0 and variance ''s'' − ''t''.
In the Poisson process, the probability distribution of ''X''_''s'' − ''X''_''t'' is a Poisson distribution with expected value λ(''s'' − ''t''), where λ > 0 is the "intensity" or "rate" of the process.
The probability distributions of the increments of any Lévy process are infinitely divisible. There is a Lévy process for each infinitely divisible probability distribution.
In any Lévy process with finite moments, the ''n''th moment $mu\_n(t)\; =\; E(X\_t^n)$ is a polynomial function of ''t''; these functions satisfy a binomial identity:
:$mu\_n(t+s)=sum\_\{k=0\}^n\; \{n\; choose\; k\}\; mu\_k(t)\; mu\_\{n-k\}(s).$
It is possible to characterise all Lévy processes by looking at their characteristic function. This leads to the 'Lévy-Khintchine representation'.

Contents

Lévy-Khintchine representation

External links

Lévy-Khintchine representation

If $X\_t$ is a Lévy process, then its characteristic function satisfies the following relation:
:
int_{mathbb{R}ackslash{0}} ig( e^{i heta x}-1 -i heta x mathbf{I}_{|x|<1}ig),W(dx) Bigg)

where $a\; in\; mathbb\{R\}$, $sigmage\; 0$ and $mathbf\{I\}$ 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-Khintchine representation of the process, are fully determined by the Lévy-Khintchine triplet $(a,sigma^2,\; W)$. So one can see that a purely continuous Lévy process is a Brownian motion with drift.

External links

★ Applebaum, David; Lévy Processes—From Probability to Finance and Quantum Groups, ''Notices of the American Mathematical Society''; vol. 51, no. 11 (December 2004).