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 and independent, and statistically identical over different time intervals of the same length. A Lévy process may thus be viewed as the continuous-time analog of a random walk.
Contents
- Mathematical definition
- Independent increments
- Stationary increments
- Infinite divisibility
- Moments
- LvyKhintchine representation
- LvyIt decomposition
- References
The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Aside from Brownian motion with drift, all other proper Lévy processes have discontinuous paths.
Mathematical definition
A stochastic process
-
X 0 = 0 almost surely - Independence of increments: For any
0 ≤ t 1 < t 2 < ⋯ < t n < ∞ ,X t 2 − X t 1 , X t 3 − X t 2 , … , X t n − X t n − 1 - Stationary increments: For any
s < t , X t − X s is equal in distribution to X t − s . - Continuity in probability: For any
ϵ > 0 andt ≥ 0 it holds thatlim h → 0 P ( | X t + h − X t | > ϵ ) = 0
If
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.
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
If
Infinite divisibility
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 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 assumption. Conversely, for each infinitely divisible probability distribution
Moments
In any Lévy process with finite moments, the nth moment
Lévy–Khintchine representation
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): If
where
A Lévy process can be seen as having three independent components: a linear drift, a Brownian motion and a superposition of independent (centered) Poisson processes with different jump sizes;
Lévy–Itō decomposition
Any Lévy process may be decomposed into the sum of a Wiener process or Brownian motion process, a linear drift and a pure jump process which captures all jumps of the original Lévy process. The latter can be thought of as a superposition of centered compound Poisson processes.This result is known as the Lévy–Itō decomposition.
Given a Lévy triplet
The process defined by
The process