Levy Process

TL;DR

Random process whose increments over non-overlapping intervals are independent and depend only on the length of the interval.
The process’s distribution does not change over time (stationary) and is independent of past values.
Sample paths may be continuous (e.g., Brownian motion) or discrete/jumpy (e.g., Poisson process).

A Lévy process is a stochastic process with the following properties:

It has independent and stationary increments: the change in the value of the process at any two different times is independent of the past and depends only on the time difference between the two times.
It has a stationary and independent distribution: the distribution of the process does not change over time and is independent of the past.
It has a continuous sample path: the value of the process can change continuously over time.

Independent and stationary increments mean that for any non-overlapping time intervals, the increments are independent of each other, and their distributions depend only on the lengths of those intervals, not on their position in time.
A stationary distribution means the probabilistic law governing the process does not change over time; being independent means the current distribution does not depend on prior values.
A continuous sample path indicates that the process’s value evolves without jumps in time. (Depending on the specific Lévy process, sample paths can also be discrete or have jumps, as shown in the examples.)

Brownian motion is a type of Lévy process describing the random motion of particles suspended in a fluid. It was first observed by Robert Brown in 1827.
Properties for Brownian motion as a Lévy process:
- The increments are independent and identically distributed.
- The distribution of the increments is normal.
- The sample path of the process is continuous.

The Poisson process is another Lévy process that describes the random arrival of events in time. It was first introduced by Siméon Denis Poisson in the 19th century; he used it to model the distribution of the number of deaths in a given time period.
Properties for the Poisson process as a Lévy process:
- The increments are independent and identically distributed.
- The distribution of the increments is Poisson with a given rate parameter.
- The sample path of the process is discrete: the number of events changes only at discrete points in time.

Modeling the random motion of particles in a fluid (Brownian motion).
Modeling random arrivals or counts of events over time, such as the number of deaths in a period (Poisson process).