Lévy process

A Lévy process is a stochastic process ${\displaystyle X=\{X_{t}:t\geq 0\}}$  that satisfies the following properties:

1. ${\displaystyle X_{0}=0\,}$  almost surely;
2. Independence of increments: For any ${\displaystyle 0\leq t_{1}<t_{2}<\cdots <t_{n}<\infty }$ , ${\displaystyle X_{t_{2}}-X_{t_{1}},X_{t_{3}}-X_{t_{2}},\dots ,X_{t_{n}}-X_{t_{n-1}}}$  are mutually independent;
3. Stationary increments: For any ${\displaystyle s<t\,}$ , ${\displaystyle X_{t}-X_{s}\,}$  is equal in distribution to ${\displaystyle X_{t-s};\,}$ 
4. Continuity in probability: For any ${\displaystyle \varepsilon >0}$  and ${\displaystyle t\geq 0}$  it holds that ${\displaystyle \lim _{h\rightarrow 0}P(|X_{t+h}-X_{t}|>\varepsilon )=0.}$ 

If ${\displaystyle X}$  is a Lévy process then one may construct a version of ${\displaystyle X}$  such that ${\displaystyle t\mapsto X_{t}}$  is almost surely right-continuous with left limits.

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 the sum 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:

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

The distribution of a Lévy process is characterized by its characteristic function, which is given by the Lévy–Khintchine formula (general for all infinitely divisible distributions):^[2]

If ${\displaystyle X=(X_{t})_{t\geq 0}}$  is a Lévy process, then its characteristic function ${\displaystyle \varphi _{X}(\theta )}$  is given by

${\displaystyle \varphi _{X}(\theta )(t):=\mathbb {E} \left[e^{i\theta X(t)}\right]=\exp {\left(t\left(ai\theta -{\frac {1}{2}}\sigma ^{2}\theta ^{2}+\int _{\mathbb {R} \setminus \{0\}}{\left(e^{i\theta x}-1-i\theta x\mathbf {1} _{|x|<1}\right)\,\Pi (dx)}\right)\right)}}$ 

where ${\displaystyle a\in \mathbb {R} }$ , ${\displaystyle \sigma \geq 0}$ , and ${\displaystyle \Pi }$  is a σ-finite measure called the Lévy measure of ${\displaystyle X}$ , satisfying the property

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

In the above, ${\displaystyle \mathbf {1} }$  is the indicator function. Because characteristic functions uniquely determine their underlying probability distributions, each Lévy process is uniquely determined by the "Lévy–Khintchine triplet" ${\displaystyle (a,\sigma ^{2},\Pi )}$ . The terms of this triplet suggest that a Lévy process can be seen as having three independent components: a linear drift, a Brownian motion, and a Lévy jump process, as described below. This immediately gives that the only (nondeterministic) continuous Lévy process is a Brownian motion with drift; similarly, every Lévy process is a semimartingale.^[3]

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 ${\displaystyle \mathbb {R} \setminus (-1,1)}$ , normalized 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.

A Lévy random field is a multi-dimensional generalization of Lévy process.^[6][7] Still more general are decomposable processes.^[8]

• Independent and identically distributed random variables
• Wiener process
• Poisson process
• Gamma process
• Markov process
• Lévy flight

• Applebaum, David (December 2004). "Lévy Processes—From Probability to Finance and Quantum Groups" (PDF). Notices of the 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..
• Kyprianou, Andreas E. (2014). Fluctuations of Lévy Processes with Applications. Introductory Lectures. Second edition. Springer. ISBN 978-3642376313..