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Concept
Lévy process
Summary
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.
A Lévy process is a stochastic process that satisfies the following properties:
almost surely;
Independence of increments: For any , are mutually independent;
Stationary increments: For any , is equal in distribution to
Continuity in probability: For any and it holds that
If is a Lévy process then one may construct a version of such that is almost surely right-continuous with left limits.
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.
Official source
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