Lévy process

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.

Independent increments
A continuous-time stochastic process assigns a random variable Xt to each point t &ge; 0 in time. In effect it is a random function of t. The increments of such a process are the differences Xs &minus; Xt between its values at different times t < s. To call the increments of a process independent means that increments Xs &minus; Xt and Xu &minus; 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 &minus; Xt depends only on the length s &minus; t of the time interval; increments with equally long time intervals are identically distributed.

In the Wiener process, the probability distribution of Xs &minus; Xt is normal with expected value 0 and variance s &minus; t.

In the Poisson process, the probability distribution of Xs &minus; Xt is a Poisson distribution with expected value &lambda;(s &minus; t), where &lambda; > 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 $$\mu_n(t) = E(X_t^n)$$ is a polynomial function of t; these functions satisfy a binomial identity:


 * $$\mu_n(t+s)=\sum_{k=0}^n {n \choose k} \mu_k(t) \mu_{n-k}(s).$$

Lévy-Khinchin representation
It is possible to characterise all Lévy processes by looking at their characteristic function. This leads to the Lévy-Khinchin representation. If $$ X_t $$ is a Lévy process, then its characteristic function satisfies the following relation:


 * $$\mathbb{E}\Big[e^{i\theta X_t} \Big] = \exp \Bigg( ait\theta - \frac{1}{2}\sigma^2t\theta^2 + t

\int_{\mathbb{R}\backslash\{0\}} \big( e^{i\theta x}-1 -i\theta x \mathbf{I}_{|x|<1}\big)\,W(dx) \Bigg) $$

where $$a \in \mathbb{R}$$, $$\sigma\ge 0$$ and $$\mathbf{I}$$ is the indicator function. The Lévy measure $$W$$ must be such that


 * $$\int_{\mathbb{R}\backslash\{0\}} \min \{ x^2, 1 \} W(dx) < \infty. $$

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-Khinchin representation of the process, are fully determined by the Lévy-Khinchin triplet $$(a,\sigma^2, W)$$. So one can see that a purely continuous Lévy process is a Brownian motion with drift.