A Lévy process is a mathematical model that describes the behavior of a certain type of random process. It is named after the French mathematician Paul Lévy, who first introduced the concept in the early 20th century.

A Lévy process is a stochastic process that has the following properties:

It has independent and stationary increments, which means that the change in the value of the process at any two different times is independent of the past and only depends on the time difference between the two times.

It has a stationary and independent distribution, which means that the distribution of the process does not change over time and is independent of the past.

It has a continuous sample path, which means that the value of the process can change continuously over time.

To understand the concept of a Lévy process better, let’s consider two examples.

Example 1: Brownian motion

Brownian motion is a type of Lévy process that describes the random motion of particles suspended in a fluid. It was first observed by the Scottish botanist Robert Brown in 1827, who noticed that the particles in a suspension of pollen grains in water seemed to move randomly.

Mathematically, Brownian motion can be modeled as a Lévy process with the following properties:

The increments of the process are independent and identically distributed, which means that the change in the position of a particle at any two different times is independent of the past and has the same distribution.

The distribution of the increments is a normal distribution, which means that the change in the position of a particle is normally distributed around its mean value.

The sample path of the process is continuous, which means that the position of a particle can change continuously over time.

Example 2: Poisson process

Another example of a Lévy process is the Poisson process, which describes the random arrival of events in time. It was first introduced by the French mathematician Siméon Denis Poisson in the 19th century, who used it to model the distribution of the number of deaths in a given time period.

Mathematically, the Poisson process can be modeled as a Lévy process with the following properties:

The increments of the process are independent and identically distributed, which means that the number of events occurring in any two different time intervals is independent of the past and has the same distribution.

The distribution of the increments is a Poisson distribution, which means that the number of events occurring in a given time interval is Poisson distributed with a given rate parameter.

The sample path of the process is discrete, which means that the number of events can only change at discrete points in time.

In summary, a Lévy process is a mathematical model that describes the behavior of a certain type of random process. It has independent and stationary increments, a stationary and independent distribution, and a continuous or discrete sample path depending on the specific type of Lévy process. Examples of Lévy processes include Brownian motion and the Poisson process.