Lévyprosessi

A Lévy process is a stochastic process with independent and stationary increments. This means that the changes in the process over non-overlapping time intervals are independent of each other, and the probability distribution of the change depends only on the length of the time interval, not on its starting point. Formally, a stochastic process X(t) is a Lévy process if it satisfies the following conditions: X(0) = 0 almost surely, the increments X(t+s) - X(t) are independent of the history of the process up to time t, and the distribution of X(t+s) - X(t) depends only on s.

Lévy processes are fundamental in probability theory and have wide-ranging applications in finance, physics, and other fields. They generalize the concept of Brownian motion, which is a specific type of Lévy process. While Brownian motion has continuous paths, many Lévy processes can exhibit jumps. This jump behavior allows them to model phenomena with sudden, discontinuous changes.

The structure of Lévy processes is deeply connected to their characteristic function, which is determined by the Lévy-Khintchine formula. This formula decomposes the process into three components: a Brownian motion part, a compound Poisson process part, and a possibly infinite activity jump part. Examples of Lévy processes include Brownian motion, the Poisson process, and the Gamma process. Their ability to model both continuous and discontinuous trajectories makes them a versatile tool for analyzing complex systems.