The Itô Equation and a General Stochastic Model for Dynamical Systems

Abstract

Activity in stochastic differential activities has been based heavily on the well known work of K. Itô who introduced stochastic integrals (Itô, 1942) to rigorously formulate the stochastic differential equation that determines Kolmogorov’s diffusion process (Kolmogorov, 1931). The Itô stochastic differential equation dx_t = f(x_t) dt + g(x_t) dw_t has long been used as a model for a dynamical system perturbed by white noise. We have assumed a basic probability space (Ω, F, µ) and that a generic point of Ω is represented by ω. The stochastic process x = (x_t (ω), 0 ≤ t < ∞) represents the “solution” of the above equation. We need not refer here to the considerable and well-documented work in stochastic calculus which has followed this pioneering work. More recently a new “input-output” point of view has developed in which (1) is regarded as a mapping taking an “input” w into an “output” which means any continuous process w is defined without the need for stochastic integration.