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:

:

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

:

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.

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