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.

Properties

Independent increments

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.

Stationary increments

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.

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 <math>\mu_n(t) = E(X_t^n)</math> is a polynomial function of t; these functions satisfy a binomial identity:

<math>\mu_n(t+s)=\sum_{k=0}^n {n \choose k} \mu_k(t) \mu_{n-k}(s).</math>

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 <math> X_t </math> is a Lévy process, then its characteristic function satisfies the following relation:

<math>\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) </math>

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

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

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 <math>(a,\sigma^2, W)</math>. So one can see that a purely continuous Lévy process is a Brownian motion with drift.

External links

• Applebaum, David (December 2004), "Lévy Processes—From Probability to Finance and Quantum Groups" (PDF), Notices of the American Mathematical Society, Providence, RI: American Mathematical Society, 51 (11): 1336–1347, ISSN 1088-9477CS1 maint: Date and year (link)