The core idea is the concept of Itô integration, where both the integrand and the integrators are stochastic processes: \[Y_t = \int_{0}^{t} H_s \, dX_s\] Here \(H\) represent an integrable stochastic process, and \(X_s\) is a Brownian motion.
The outcome of the above integration is yet another stochastic process with time parameter \(t\). This can also be expressed in differential form: \[dY = H \, dX\] which is analogous to: \[Y - Y_0 = H \cdot X \]
Itō calculus plays a crucial role in understanding and modeling systems involving randomness, making it a fundamental tool in fields like mathematical finance and stochastic differential equations.
In this simulation, the blue curve represents the Brownian motion \(B_t\), and the red curve represents the Itō integral \(Y_t(B)\) of the Brownian motion with respect to itself.