Lévyprosesseilla

Lévyprosesseilla, also known as Lévy processes, are a class of stochastic processes that generalize Brownian motion. They are characterized by having stationary and independent increments. This means that the change in the process over any time interval depends only on the length of the interval, not on when the interval occurs, and the changes over non-overlapping intervals are independent of each other.

A key property of Lévy processes is that their increments are infinitely divisible. This implies that the distribution of the increment over any time interval can be represented as the distribution of a sum of independent and identically distributed random variables. This infinite divisibility is a fundamental concept and distinguishes Lévy processes from processes with only independent increments.

Brownian motion is a well-known example of a Lévy process, but Lévy processes encompass a much broader family. They can exhibit jumps, which are discontinuities in the sample paths. The presence and nature of these jumps are determined by the specific Lévy measure associated with the process. Lévy processes are widely used in various fields, including finance, physics, and engineering, to model phenomena that involve randomness and can exhibit sudden changes or jumps. The mathematical framework for Lévy processes is deeply rooted in probability theory and Fourier analysis, particularly through the use of characteristic functions.