Lévy process

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In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

The most well known examples of Lévy processes are Brownian motion and the Poisson process. Aside from Brownian motion with drift, all other Lévy processes, except the deterministic case, have discontinuous paths.

Mathematical definition

A stochastic process $X=\{X_{t}:t\geq 0\}$ is said to be a Lévy process if it satisfies the following properties:

$X_{0}=0\,$ almost surely
Independence of 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 $X_{{t-s}}.\,$
Continuity in probability: For any $\epsilon >0$ and $t\geq 0$ it holds that $\lim _{{h\rightarrow 0}}P(|X_{{t+h}}-X_{t}|>\epsilon )=0$

If $X$ is a Lévy process then one may construct a version of $X$ such that $t\mapsto X_{t}$ 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_t − X_s depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed.

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

If $X$ is the 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.

Infinite divisibility

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 assumption. Conversely, for each infinitely divisible probability distribution $F$ , there is a Lévy process $X$ such that the law of $X_{1}$ is given by $F$ .

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–Khintchin representation

The distribution of a Lévy process is characterized by its characteristic function, which is given by the Lévy–Khinchine formula: If $(X_{t})_{{t\geq 0}}$ is a Lévy process, then its characteristic function is given by

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

where $a\in {\mathbb {R}}$ , $\sigma \geq 0$ , ${\mathbf {I}}$ is the indicator function and $\Pi$ is a sigma-finite measure called the Lévy measure of $X$ , which satisfies the property

$\int _{{{\mathbb {R}}\backslash \{0\}}}\min\{x^{2},1\}\Pi (dx)<\infty .$

A Lévy process can be seen as having three independent components: a linear drift, a Brownian motion and a superposition of independent (centered) Poisson processes with different jump sizes; $\Pi (dx)$ represents the rate of arrival (intensity) of the Poisson process with jump of size $x$ . These three components, and thus the Lévy–Khintchine representation, are fully determined by the Lévy–Khintchine triplet $(a,\sigma ^{2},\Pi )$ . In particular, the only (nondeterministic) continuous Lévy process is a Brownian motion with drift.

Lévy–Itō decomposition

Any Lévy process may be decomposed into the sum of a Brownian motion, a linear drift and a pure jump process which captures all jumps of the original Lévy process. The latter can be thought of as a superposition of centered compound Poisson processes.This result is known as the Lévy–Itō decomposition.

Given a Lévy triplet $(a,\sigma ^{2},\Pi )$ 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 then a Lévy process with triplet $(a,\sigma ^{2},\Pi )$ .

The process $X^{{(3)}}$ can be further decomposed as a sum of two independent processes the first pure jump zero mean martingale of jumps less than $1$ In absolute value and the second a compound Poisson process describing the jumps bigger than one in absolute value.

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.
Cont, Rama; Tankov, Peter (2003). Financial Modeling with Jump Processes. CRC Press. ISBN 978-1584884132. .
Sato, Ken-Iti (2011). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. ISBN 978-0521553025. .

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