In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are 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 processes.[1]
X=\{Xt:t\geq0\}
X0=0
0\leqt1<t2< … <tn<infty
| X | |
| t2 |
| -X | |
| t1 |
,
| X | |
| t3 |
| -X | |
| t2 |
| ,...,X | |
| tn |
| -X | |
| tn-1 |
s<t
Xt-Xs
Xt-s;
\varepsilon>0
t\ge0
\limh → P(|Xt+h-Xt|>\varepsilon)=0.
If
X
X
t\mapstoXt
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.
See main article: Stationary increments. To call the increments stationary means that the probability distribution of any increment Xt − Xs depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed.
If
X
If
X
If
X
f(x;t)={1\over\pi}\left[{t\overx2+t2}\right]
The distribution of a Lévy process has the property of infinite divisibility: given any integer n, the 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
F
X
X1
F
In any Lévy process with finite moments, the nth moment
\mun(t)=
| n) | |
| E(X | |
| t |
\mun(t+s)=\sum
| n | |
| k=0 |
{n\choosek}\muk(t)\mun-k(s).
The distribution of a Lévy process is characterized by its characteristic function, which is given by the Lévy–Khintchine formula (general for all infinitely divisible distributions):[2]
Ifis a Lévy process, then its characteristic functionX=(Xt)t\geq
is given by\varphiX(\theta)
}\right)\right)}where\varphiX(\theta)(t):=E\left[ei\theta\right]=\exp{\left(t\left(ai\theta-
1 2 \sigma2\theta2+\int\R\setminus\{0\
,a\inR
, and\sigma\ge0
is a -finite measure called the Lévy measure of\Pi
, satisfying the propertyX
} < \infty.\int\R\setminus\{0\
In the above,
1
(a,\sigma2,\Pi)
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= | \Pi|\R\setminus(-1,1) |
| \Pi(\R\setminus(-1,1)) |
\Pi
\R\setminus(-1,1)
\mu=\Pi|(-1,1)\setminus\{0\
\int\R\setminus\{0\
The former is the characteristic function of a compound Poisson process with intensity
\Pi(\R\setminus(-1,1))
\nu
1
\int\R{|x|\mu(dx)}<infty
X
Xt=\sigmaBt+at+Yt+Zt,t\geq0,
Y
1
Zt
A Lévy random field is a multi-dimensional generalization of Lévy process.[5] Still more general are decomposable processes.[6]