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Lévy process
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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.
class="link1" onMouseover='showByLink("m2854886",this)' onMouseout='hide("m2854886")'href="http://www.absoluteastronomy.com/topics/Stochastic_process">stochastic process is said to be a Lévy process if,
- almost surely
- For any , are independent
- For any , is equal in distribution to
- is almost surely right continuous with left limits.
ntinuous-time stochastic process assigns a random variable Xt to each point t = 0 in time.
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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.
Mathematical Definition
A stochastic process is said to be a Lévy process if,
- almost surely
- For any , are independent
- For any , is equal in distribution to
- is almost surely right continuous with left limits.
Properties
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 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 ?(s − t), where ? > 0 is the "intensity" or "rate" of the process.
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.
Moments
In any Lévy process with finite moments, the nth moment is a polynomial function of t; these functions satisfy a binomial identity:
Lévy–Khintchine representation
It is possible to characterise all Lévy processes by looking at their characteristic function. This leads to the Lévy–Khintchine representation. If is a Lévy process, then its characteristic function satisfies the following relation:
where , and is the indicator function. The Lévy measure 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 . So one can see that a purely continuous Lévy process is a Brownian motion with drift.
Lévy–Ito decomposition
We can also construct a Lévy process from any given characteristic function of the form given in the Lévy–Khintchine representation. Given a Lévy triplet there exists three independent Lévy processes, which lie in the same probability space, , , such that is a Brownian motion with drift, is a compound Poisson process and is a square integrable pure jump martingale that almost surely has a countable number of jumps on a finite interval. The process defined by is a Lévy process with triplet .
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