Lévyprocesser

Lévy processes are a class of stochastic processes that are fundamental in probability theory and have applications in various fields such as finance, physics, and biology. A Lévy process is characterized by three key properties: stationary increments, independent increments, and almost sure right continuity with left limits (càdlàg trajectories). Stationary increments mean that the distribution of the process's change over a time interval depends only on the length of the interval, not on its starting point. Independent increments imply that the changes in the process over disjoint time intervals are independent random variables. The càdlàg property ensures that the paths of the process have no jumps upwards and are continuous from the right.

The simplest and most well-known example of a Lévy process is the Brownian motion, also known as the Wiener process. Brownian motion has continuous trajectories and its increments are normally distributed. However, Lévy processes can exhibit more complex behavior, including jumps. The Lévy-Khintchine representation provides a canonical decomposition of any Lévy process based on its drift, a diffusion component (which corresponds to Brownian motion), and a jump measure that describes the frequency and size of jumps.

Other important examples of Lévy processes include the Poisson process, which models events occurring randomly over time, and compound Poisson processes, where jumps have random magnitudes. More general Lévy processes can have infinitely many small jumps or a finite number of large jumps. The study of Lévy processes is crucial for understanding phenomena with random fluctuations and unpredictable jumps, making them indispensable tools in mathematical modeling.