# Lévy Processes (DRAFT)

```{admonition} Part of a series: Stochastic fundamentals.
Follow reading [here](fundamentals.md)
```

An important class of [stochastic processes](stoch_process.md) is given by Lévy processes. They provide a flexible framework for processes with stationary and independent increments that allows for continuous processes as well as processes with jump discontinuities. For example, it encompasses well-known processes like Brownian motion and Poisson processes. 

## Definition
For the rest of this article, it is assumed that the underlying [probability space](probab_space.md) is denoted by $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathbb{F}=(\mathcal{F}_t)_{t\ge0}$ is a filtration that is assumed to be right-continuous and complete.

An adapted stochastic process $X=(X_t)_{t\ge0}$ on $\mathbb{R}$ is a (one-dimensional) Lévy process if the following conditions are satisfied:

1. $X_0=0~\mathbb{P}$-almost surely.
2. For any $n\in\mathbb{N}$ and $0\le t_0<t_1<\dots<t_n<\infty$, the random variables $X_{t_0},X_{t_1}-X_{t_0},\dots,X_{t_n}-X_{t_{n-1}}$ are independent.
3. $X_{s+t}-X_s\stackrel{d}{=}X_t$ for $0\le s,t<\infty$.
4. $X$ is continuous in probability, i.e. for every $t\ge0$ and $\varepsilon>0$, it holds that 

$$
\begin{align*}
\lim_{s\to t}\mathbb{P}(|X_s-X_t|>\varepsilon)=0.
\end{align*}
$$

Condition 2 is called the "independent increments property". It means that the behaviour of a Lévy process is independent along different time intervals. The "stationary increments property" 3 implies that, probabilistically, a process behaves equally in every time interval with the same length {cite}`Applebaum2009`.

## Properties 
In this section, some characteristic properties of Lévy processes are collected {cite}`Protter2004`.

* The characteristic function of a Lévy process $X$ is determined by the so-called Lévy-Khintchine formula. For $t\ge0$, it states that

  $$
  \begin{align*}
  \phi_{X_t}(u)=\mathbb{E}\left[\mathrm{e}^{-\mathrm{i}uX_t}\right]=\mathrm{e}^{-t\psi(u)},\quad u\in\mathbb{R},
  \end{align*}
  $$

  where the function $\psi\colon\mathbb{R}\to\mathbb{C}$ is of the form

  $$
  \begin{align*}
  \psi(u)=\frac{\sigma^2}{2}u^2-\mathrm{i}\gamma u+\int_{\mathbb{R}\setminus\{0\}}\left(1-\mathrm{e}^{-\mathrm{i}ux}+\mathrm{i}ux\mathbb{1}_{\{|x|<1\}}(x)\right)\nu(\mathrm{d}x),
  \end{align*}
  $$

  where $\sigma\ge0,~\gamma\in\mathbb{R}$ and $\nu$ is a measure on $\mathbb{R}\setminus\{0\}$ satisfying

  $$
  \begin{align*}
  \int_{\mathbb{R}\setminus\{0\}}(1\wedge x^2)\nu(\mathrm{d}x)<\infty.
  \end{align*}
  $$

  A Lévy process is uniquely characterised by its corresponding parameters $\sigma,~\gamma$ and $\nu$. In particular, $\gamma$ describes the drift of such a process, $\sigma$ represents the Brownian motion component and $\nu$ is the Lévy measure determining jump intensities. The triplet $(\gamma,\sigma^2,\nu)$ is referred to as the generating triplet.

* Every Lévy process $X$ can assumed to be càdlàg (i.e. the paths $t\mapsto X_t$ are right-continuous $\mathbb{P}$-a.s. and the left limits exist $\mathbb{P}$-a.s.).[^1]

* If there exists a constant $C>0$ such that $\sup_{0\le t<\infty}|\Delta X_t|<C$, where $\Delta X_t=X_t-X_{t-}$ with $X_{t-}=\lim_{s\uparrow t}X_s$, then $\mathbb{E}[|X_t|^n]<\infty$ for every $t\ge0$ and $n\in\mathbb{N}$.

* Lévy processes are examples of [Markov processes](markov_process.md) and semimartingales.

## Examples
By concretely determining distributions for the stationary increments property 3, different kinds of Lévy processes arise {cite}`Applebaum2009,Bass2011`). Corresponding paths are depicted in {numref}`fig_processes`.

### Brownian Motion
One of the most prominent stochastic processes is Brownian motion. A process $B=(B_t)_{t\ge0}$ is a Brownian motion if it is a Lévy process and if $B_t\sim\mathcal{N}(0,t),~t\ge0$. It can also be seen as an [example of a Markov process](markov_ex.md).

The generating triplet of a Brownian motion is given by $(\gamma,\sigma^2,\nu)=(0,1,0)$. In the context of Lévy processes, a Brownian motion $(B_t)_{t\ge0}$ exhibits some interesting path properties:

* The paths $t\mapsto B_t$ are continuous $\mathbb{P}$-a.s.
* The paths $t\mapsto B_t$ are $\mathbb{P}$-a.s. nowhere differentiable.

The continuity of the paths is a particular strong property since the paths of a Lévy process are in general only càdlàg.


### Poisson Process
A Poisson process is a typical pure jump process whose name stems from the fact that the increments are Poisson distributed. More formally, a process $N=(N_t)_{t\ge0}$ is said to be a Poisson process with parameter $\lambda>0$ if it is a Lévy process and if $N_t\sim Pois(\lambda t),~t>0$. [Here](markov_ex.md), Poisson processes are discussed as examples of Markov processes.

The generating triplet is $(\gamma,\sigma^2,\nu)=(0,0,\lambda\delta_1)$, where $\delta_x$ denotes the Dirac measure on $\mathbb{R}$ for given $x\in\mathbb{R}$. Furthermore, it can be shown that the paths of a Poisson process are increasing and constant except for jumps of size $1~\mathbb{P}$-a.s.

### Compound Poisson Process
Compound Poisson processes are also pure jump processes, but they generalize Poisson processes by enabling processes to have arbitrary jump sizes.

For a mathematical definition, let $N$ be a Poisson process with parameter $\lambda>0$ and let $(Z_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables taking values in $\mathbb{R}$ with common law $\mu_Z$, where $\mu_Z$ is a distribution on $\mathbb{R}$ with $\mu_Z(\{0\})=0$. It is supposed that $N$ is independent of all $Z_k$'s. A compound Poisson process $Y=(Y_t)_{t\ge0}$ is then defined as

$$
\begin{align*}
Y_t=\sum_{k=1}^{N_t}Z_k,\quad t\ge0.
\end{align*}
$$

The generating triplet of such a compound Poisson process is

$$
\begin{align*}
(\gamma,\sigma^2,\nu)=\left(\lambda\int_{\{|x|<1\}}x\mu_Z(\mathrm{d}x),0,\lambda\mu_Z\right).
\end{align*}
$$

As already mentioned, the definition of a compound Poisson processes allows for random jump sizes. They can take all values supported by the distribution $\mu_Z$. If $\mu_Z=\lambda\delta_1$, for $\lambda>0$, a compound Poisson process reduces to a Poisson process with parameter $\lambda$.


```{figure} figures/stochastic/levy_processes.png
:alt: Trajectories of Lévy processes
:align: center
:class: bg-primary mb-1
:name: fig_processes
Paths of a Brownian motion, a Poisson process and a compound Poisson process
```


## Literature
```{bibliography}
:filter: docname in docnames  
```

## Authors
Julian Wäsche

## Contributors
Philipp Böttcher

[^1]: More precisely, for any Lévy process, there exists a unique modification which is càdlàg $\mathbb{P}$-a.s. and which is also a Lévy process.