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. They are a stochastic analog of independent and identically-distributed random variables, and the most well-known examples are the Wiener process and the Poisson process.

Definition

A stochastic process $X=\{X_{t}:t\geq 0\}$ is said to be a Lévy process if,

$X_{0}=0\,$ almost surely
Independent increments: For any $0\leq t_{1}<t_{2}<\cdots <t_{n}<\infty$ , $X_{t_{2}}-X_{t_{1}},X_{t_{3}}-X_{t_{2}},\dots ,X_{t_{n}}-X_{t_{n-1}}$ are independent
Stationary increments: For any $s<t\,$ , $X_{t}-X_{s}\,$ is equal in distribution to $X_{t-s}\,$
$t\mapsto X_{t}$ is almost surely right continuous with left limits.

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 (homogeneous) 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

Lévy processes correspond to infinitely divisible probability distributions:

The probability distributions of the increments of any Lévy process are infinitely divisible, since the increment of length t is the sum of n increments of length t/n, which are i.i.d. by assumption (independent increments and stationarity).
Conversely, there is a Lévy process for each infinitely divisible probability distribution: given such a distribution D, multiples and dividing define a stochastic process for positive rational time, defining it as a Dirac delta distribution for time 0 defines it for time 0, and taking limits defines it for real time. Independent increments and stationarity follow by assumption of divisibility, though one must check continuity and that taking limits gives a well-defined function for irrational time.

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–Khintchine representation

It is possible to characterise all Lévy processes by looking at their characteristic function. This leads to the Lévy–Khintchine 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 ^{2}t\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 $a\in \mathbb {R}$ , $\sigma \geq 0$ 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–Khintchine representation of the process, are fully determined by the Lévy–Khintchine triplet $(a,\sigma ^{2},W)$ . So one can see that a purely continuous Lévy process is a Brownian motion with drift.

Lévy–Itō decomposition

We can also construct a Lévy process from any given characteristic function of the form given in the Lévy–Khintchine representation. This expression corresponds to the decomposition of a measure in Lebesgue's decomposition theorem: the drift and diffusion are the absolutely continuous part, while the measure W is the singular measure.

Given a Lévy triplet $(a,\sigma ^{2},W)$ there exists three independent Lévy processes, which lie in the same probability space, $X^{(1)}$ , $X^{(2)}$ , $X^{(3)}$ such that:

$X^{(1)}$ is a Brownian motion with drift, corresponding to the absolutely continuous part of a measure and capturing the drift a and diffusion $\sigma ^{2}$ ;
$X^{(2)}$ is a compound Poisson process, corresponding to the pure point part of the singular measure W;
$X^{(3)}$ is a square integrable pure jump martingale that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part of the singular measure W.

The process defined by $X=X^{(1)}+X^{(2)}+X^{(3)}$ is a Lévy process with triplet $(a,\sigma ^{2},W)$ .

External links

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