Wiener meets Kolmogorov

Abstract

Stochastic processes are ubiquitous and the stochastic differential equation is a formal setting to analyse dynamic circuits in noisy environments. Initially, Norbert Wiener developed stochastic integral to prove the nondifferentiability of the Brownian motion with probability one. Later, it was utilized by Kiyoshi ItΣ to construct stochastic differential rules. This paper discusses two Theorems. The proof of the first Theorem reveals a connection between the stochastic differential rules of the Wiener process and the Kolmogorov forward equation. The second Theorem demonstrates the application of the Kolmogorov forward and backward equations to achieve the non-linear filtering of a non-linear dynamic circuit with embedded stochasticity. The problem of embedding stochasticity into circuits and systems and achieving the nonlinear filtering of electronic circuits becomes a potential problem. That will refine the unified theory as well as algorithmic procedures for networked control. This paper demonstrates the beauty, power and universality of the Wiener process and related results, i.e. the Kolmogorov forward and backward equations, in non-linear filtering of stochastic control problems.