Lévy 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_t − X_s depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed.

If ${\displaystyle X}$ is a Wiener process, the probability distribution of X_t − X_s is normal with expected value 0 and variance t − s.

If ${\displaystyle X}$ is a Poisson process, the probability distribution of X_t − X_s is a Poisson distribution with expected value λ(t − s), where λ > 0 is the "intensity" or "rate" of the process.

If ${\displaystyle X}$ is a Cauchy process, the probability distribution of X_t − X_s is a Cauchy distribution with density ${\displaystyle f(x;t)={1 \over \pi }\left[{t \over x^{2}+t^{2}}\right]}$.

The distribution of a Lévy process has the property of infinite divisibility: given any integer n, the law of a Lévy process at time t can be represented as the law of n independent random variables, which are precisely the increments of the Lévy process over time intervals of length t/n, which are independent and identically distributed by assumptions 2 and 3. Conversely, for each infinitely divisible probability distribution ${\displaystyle F}$, there is a Lévy process ${\displaystyle X}$ such that the law of ${\displaystyle X_{1}}$ is given by ${\displaystyle F}$.

In any Lévy process with finite moments, the nth moment ${\displaystyle \mu _{n}(t)=E(X_{t}^{n})}$, is a polynomial function of t; these functions satisfy a binomial identity:

${\displaystyle \mu _{n}(t+s)=\sum _{k=0}^{n}{n \choose k}\mu _{k}(t)\mu _{n-k}(s).}$

Because the characteristic functions of independent random variables multiply, the Lévy–Khintchine theorem suggests that every Lévy process is the sum of Brownian motion with drift and another independent random variable, a Lévy jump process. The Lévy–Itô decomposition describes the latter as a (stochastic) sum of independent Poisson random variables.

Let ${\displaystyle \nu ={\frac {\Pi |_{\mathbb {R} \setminus (-1,1)}}{\Pi (\mathbb {R} \setminus (-1,1))}}}$— that is, the restriction of ${\displaystyle \Pi }$ to ${\displaystyle \mathbb {R} \setminus (-1,1)}$, renormalized to be a probability measure; similarly, let ${\displaystyle \mu =\Pi |_{(-1,1)\setminus \{0\}}}$ (but do not rescale). Then

${\displaystyle \int _{\mathbb {R} \setminus \{0\}}{\left(e^{i\theta x}-1-i\theta x\mathbf {1} _{|x|<1}\right)\,\Pi (dx)}=\Pi (\mathbb {R} \setminus (-1,1))\int _{\mathbb {R} }{(e^{i\theta x}-1)\,\nu (dx)}+\int _{\mathbb {R} }{(e^{i\theta x}-1-i\theta x)\,\mu (dx)}.}$

The former is the characteristic function of a compound Poisson process with intensity ${\displaystyle \Pi (\mathbb {R} \setminus (-1,1))}$ and child distribution ${\displaystyle \nu }$. The latter is that of a compensated generalized Poisson process (CGPP): a process with countably many jump discontinuities on every interval a.s., but such that those discontinuities are of magnitude less than ${\displaystyle 1}$. If ${\displaystyle \int _{\mathbb {R} }{|x|\,\mu (dx)}<\infty }$, then the CGPP is a pure jump process.[4][5] Therefore in terms of processes one may decompose ${\displaystyle X}$ in the following way

${\displaystyle X_{t}=\sigma B_{t}+at+Y_{t}+Z_{t},t\geq 0,}$

where ${\displaystyle Y}$ is the compound Poisson process with jumps larger than ${\displaystyle 1}$ in absolute value and ${\displaystyle Z_{t}}$ is the aforementioned compensated generalized Poisson process which is also a zero-mean martingale.