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In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that has "stationary independent increments" -- this phrase will be explained below. The most well-known examples are the Wiener process and the Poisson process.
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. 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 Xt is normal with expected value 0 and variance t.
In the Poisson process, the probability distribution of Xt is a Poisson distribution with expected value λt, where λ > 0 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.
In any Lévy process with finite moments, the nth moment mn(t) = E(Xtn) is a polynomial function of t; these functions satisfy a binomial identity:
- <math>m_n(t+s)=\sum_{k=0}^n {n \choose k} m_k(t) m_{n-k}(s).<math>
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