Lévyprosessien

Lévyprosessien, also known as Lévy processes, are a class of stochastic processes that are of fundamental importance in probability theory and have numerous applications in fields such as finance, physics, and signal processing. A Lévy process is a stochastic process with stationary and independent increments. Stationary increments mean that the distribution of the increment of the process over any time interval depends only on the length of the interval, not on its starting point. Independent increments imply that the increments over disjoint time intervals are independent random variables.

A key characteristic of Lévy processes is their connection to infinitely divisible distributions. A probability distribution is infinitely divisible if it can be expressed as the distribution of the sum of an arbitrary number of independent and identically distributed random variables. The increments of a Lévy process are always infinitely divisible.

The simplest and most well-known example of a Lévy process is Brownian motion, also known as the Wiener process. Brownian motion has continuous paths and its increments are normally distributed. However, Lévy processes can exhibit much more complex behavior, including jumps. Processes with jumps are often referred to as jump processes.

The general structure of a Lévy process is described by the Lévy-Khintchine formula, which provides a representation of the characteristic function of the process. This formula highlights the three main components that characterize a Lévy process: a drift term, a diffusion term (associated with Brownian motion), and a jump measure that governs the frequency and size of the jumps. The study of Lévy processes involves understanding their probabilistic properties, their simulation, and their application to modeling real-world phenomena where sudden, discontinuous changes are observed.