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. It is 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,

1. $X_0=0 \,$ almost surely
2. 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
3. Stationary increments: For any $s<t \,$, $X_t-X_s \,$ is equal in distribution to $X_{t-s} \,$
4. $t \mapsto X_t$ 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

See also: Stationary process

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–Khinchine representation

It is possible to characterise all Lévy processes by looking at their characteristic function. This leads to the Lévy–Khinchine representation. If $X_t$ is a Lévy process, then its characteristic function satisfies the following relation:

$\mathbb{E}\Big[e^{i\theta X_t} \Big] = \exp \Bigg( ait\theta - \frac{1}{2}\sigma^2t\theta^2 + t \int_{\mathbb{R}\backslash\{0\}} \big( e^{i\theta x}-1 -i\theta x \mathbf{I}_{|x|<1}\big)\,W(dx) \Bigg)$

where $a \in \mathbb{R}$, $\sigma\ge 0$ 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 having 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 $(a,\sigma^2, W)$.

Constructing a stochastic probability measure

Consider a random process; $X_{\cdot}:\mathbb{R}_{\geq 0}\to G$ with independent increments, where the random values occur in, say, a second countable locally compact abelian group $G$.

Let $\mu_{0},\mu_{s,t}:\sigma(\dot{G})\to[0,1]$ for $s<t\in\mathbb{R}_{\geq 0}$ denote the (borel regular) probability measures on the initial position and the increments. Now for $0=t_{0}< t_{1}< t_{2}<\ldots <t_{n}<\infty$ let

$\mu_{t_{0},t_{1},\ldots,t_{n}}(A)= \int {}_{g_{0},d_{0}\ldots,d_{n-1}}~\chi_{A}(g_{0}+\Sigma_{i<k}d_{i})_{k\leq n}~d\mu_{0}\times\Pi_{i<n}\mu_{t_{i},t_{t+1}}$.

These define bona fide probability measures which, by the properties of the process, compute appropriate probabilities for properties of paths depending on only finitely many times.

Now correspond to these measures continuous linear operators

$\Lambda_{t_{0},t_{1},\ldots,t_{n}}:C_{\infty}(\Pi_{t_{i},i\leq n}\dot{G},\mathbb{R})\to\mathbb{R}$

in the obvious way. Then, for any countable set of times (for ease consider the rationals $\mathbb{Q}$) define a linear functional,

$E:C_{\infty}(\Pi_{q\in\mathbb{Q},\geq 0}\dot{G},\mathbb{R})\to\mathbb{R}$

as follows. If $f$ depends only on finitely many times, say $f(\vec{g})=\phi(g_{t_{0}},g_{t_{1}},\ldots,g_{t_{n}})$ where without loss of generality $0=t_{0}< t_{1}< t_{2}<\ldots <t_{n}<\infty$, let

$E(f)=\Lambda_{t_{0},t_{1},\ldots,t_{n}}(\phi).$

It is straightforward to see that this is well-defined and linear.^{[citation needed]} Moreover it is clearly a positive, bounded operator with $\|E\|=1$ since $E(1)=1$. By Stone-Weierstrass, $E$ extends (uniquely) to a (linear) continuous (positive) operator (with norm 1) on its domain. By the Riesz representation theorem, this in turn gives rise to a (unique) (borel regular) probability measure,

$\mu_{\mathbb{Q}}:\sigma(\Pi_{q\in\mathbb{Q},\geq 0}\dot{G})\to[0,1].$

Precisely, this measure is the unique one satisfying the condition that

$\mu_{\mathbb{Q}}(\pi_{t_{0},t_{1},\ldots,t_{n}}^{-1}A)=\mu_{t_{0},t_{1},\ldots,t_{n}}(A)$

for any $0=t_{0}<t_{1}<\ldots<t_{n},\in\mathbb{Q}$, $A\in\sigma(\Pi_{t_{0},t_{1},\ldots,t_{n}}\dot{G})$.

Whereas initially we knew the probability distributions of a path at given times or over time increments, and thus could talk about local properties of the paths in the stochastic process, the constructed measure above allows us to attach a probability distribution to (almost) the full path space, and thus enables us to talk about global properties. Roughly we are justified (and compelled to) thinking of the measure as though $\mu_{\mathbb{Q}}(A)$ calculates "the probability" that a path occurs in $A$ (when projected onto the times $\mathbb{Q}$).

As an example of our new ability to talk about global properties, we have that "almost every path is left continuous", if and only if, for every countable sequence of times $t_{n}\rightarrow\breve{t}^{-}$, letting $Q=\{t_{n},\breve{t}:n\in\mathbb{N}\}$, we have that for all $\mu_{Q}$-almost-everywhere $\vec{x}\in\Pi_{t\in Q}\dot{G}$ then $x_{t_{n}}$ converges / converges to $x_{\breve{t}}$. This makes sense, as it can be shown that

• $f:\mathbb{R}\to Y$ has left limits/is left cts if and only if $f:\mathbb{R}\to Y$ has limits/is cts under the topology on $\mathbb{R}$ generated by $(q,r],q,r\in\mathbb{Q}$; and
• if $X$ is second countable then $f:X\to Y$ has limits/is cts if and only if $f(x_{n})$ converges / converges to $f(\breve{x})$ whenever $x_{n}\rightarrow \breve{x}$.

Verifying how global properties of paths over the real line can be translated into properties considering only countably many times, can be a little tricky. There is no escaping this. Fortunately, the problem of having to change the countable set of times over which the measure is based can be prevented. If we consider a countable dense subset, $Q$, of the reals (e.g. the rationals), we may apply knowledge of the distribution on the increments together with the stochastic measure $\mu_{Q}$ to check these global properties. For example, in the case of the Wiener process, we are able to check that almost every path is (i) everywhere cts; (ii) has continuity modulus $k\sqrt{\delta\log(\delta)}$ (Lévy); and thus (iii) is nowhere differentiable.

See also

• Independent and identically distributed random variables

References

• 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 1088-9477, http://www.ams.org/notices/200411/fea-applebaum.pdf 

• Y. T. Su, K. T. Wong & K.-P. Ho, "Linear MMSE Estimation of Large-Magnitude Symmetric Levy-Process Phase-Noise", IEEE Transactions on Communications.