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 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 X_t 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 X_s − X_t between its values at different times t < s. To call the increments of a process independent means that increments X_s − X_t and X_u − X_v 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 X_s − X_t 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 X_s − X_t is normal with expected value 0 and variance s − t.

In the Poisson process, the probability distribution of X_s − X_t is a Poisson distribution with expected value λ(s − t), 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 m_n(t) = E(X_tⁿ) is a polynomial function of t; these functions satisfy a binomial identity:

$m_n(t+s)=\sum_{k=0}^n {n \choose k} m_k(t) m_{n-k}(s).$

Categories: Stochastic processes