Lévyprosesseissa

Lévyprosesseissa, often translated as Lévy processes, are a fundamental class of stochastic processes that generalize the concept of Brownian motion. They are characterized by their stationary and independent increments, meaning that the change in the process over any time interval depends only on the length of the interval and is independent of changes in other, non-overlapping intervals. Furthermore, the increments are assumed to follow a Lévy distribution.

A key feature of Lévy processes is their ability to model phenomena that exhibit jumps. Unlike Brownian motion, which has continuous paths, Lévy processes can have discontinuous paths, reflecting sudden, unpredictable changes. This makes them particularly useful in fields like finance, where they can model stock price movements that experience sharp drops or rises.

Mathematically, a stochastic process $X(t)$ is a Lévy process if $X(0) = 0$ almost surely, its increments $X(t+s) - X(t)$ are independent of the past history $X(u)$ for $u \le t$, and the distribution of $X(t+s) - X(t)$ depends only on $s$. The characteristic function of a Lévy process is described by the Lévy-Khintchine formula, which decomposes the process into a Brownian component, a drift term, and a jump component. This decomposition highlights the rich structure and versatility of these processes in modeling a wide range of random phenomena.