Itôdiffusjon

Itô diffusion, also known as Itô process, is a type of stochastic process named after the Japanese mathematician Kiyoshi Itô. It is a continuous-time stochastic process that models the evolution of a system subject to random fluctuations. The Itô diffusion process is characterized by its infinitesimal drift and diffusion coefficients, which describe the deterministic and random components of the process, respectively.

The Itô diffusion process is defined by the stochastic differential equation (SDE) of the form:

dX(t) = μ(X(t), t) dt + σ(X(t), t) dW(t)

where X(t) is the process at time t, μ(X(t), t) is the drift coefficient, σ(X(t), t) is the diffusion coefficient, and W(t) is a Wiener process (also known as Brownian motion). The drift coefficient determines the average rate of change of the process, while the diffusion coefficient determines the volatility or randomness of the process.

Itô diffusion processes are widely used in various fields, including finance, physics, and engineering, to model systems with random fluctuations. They are particularly useful in the study of financial derivatives, where they are used to model the evolution of asset prices. The Itô calculus, developed by Itô, provides a set of rules for manipulating Itô diffusion processes and is essential for solving SDEs and calculating expected values of functionals of Itô diffusion processes.