LévyItô

LévyItô refers to a class of stochastic processes that combine properties of Lévy processes and Itô processes. A Lévy process is a stochastic process with stationary and independent increments, and whose paths are almost surely right-continuous with left limits. Examples include Brownian motion and the Poisson process. An Itô process is a stochastic process that can be represented as a stochastic differential equation of the form dX_t = a(t, X_t)dt + b(t, X_t)dW_t, where W_t is a standard Brownian motion.

LévyItô processes generalize Itô processes by allowing for jumps. They are often defined as solutions to stochastic differential equations involving both a Brownian motion term and a general Lévy process term. This means that while they possess the continuous diffusion component characteristic of Itô processes, they can also exhibit sudden, discontinuous jumps, a feature inherited from Lévy processes. The mathematical framework for studying these processes relies on the theory of stochastic calculus, particularly Itô calculus, extended to accommodate jumps.

The study of LévyItô processes is crucial in various fields, including finance, physics, and engineering, where phenomena often involve both continuous fluctuations and abrupt changes. For instance, in financial modeling, LévyItô processes are used to describe asset prices that experience both gradual movements and sudden market crashes or rallies. Their ability to model both diffusion and jumps makes them a powerful tool for capturing complex real-world dynamics that simpler processes might not adequately represent.