In probability theory, a Lévy process, named after the French mathematician Paul Pierre Lévy is any continuous-time stochastic process that starts at 0, admits càdlàg (right-continuous with left limits) modification and has “stationary independent increments”. The most well-known examples are the Wiener process and the Poisson process.
A continuous-time stochastic process assigns a random variable to each point in time. In effect it is a random function of . The increments of such a process are the differences between its values at different times . To call the increments of a process independent means that increments and 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 depends only on the length of the time interval; increments with equally long time intervals are identically distributed.
In the Poisson process, the probability distribution of is a Poisson distribution with expected value , where 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.
It is possible to characterise all Lévy processes by looking at their characteristic function.
This entry was adapted from the Wikipedia article http://en.wikipedia.org/wiki/Levy_processLévy process as of January 25, 2007.
|Date of creation||2013-03-22 16:37:19|
|Last modified on||2013-03-22 16:37:19|
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