Lévy process

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" &ge; 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 &lambda;("s" − "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 "n"th 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 heta X_t} Big] = exp Bigg( ait heta - frac{1}{2}sigma^2t heta^2 + t int_{mathbb{R}ackslash{0 ig( e^{i heta x}-1 -i heta x mathbf{I}_{|x|<1}ig),W(dx) Bigg)

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

:int_{mathbb{R}ackslash{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.

External links

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