Convergence and Confidence Intervals of Sample Paths of a Stochastic Differential Equation

Abstract

For a linear scalar stochastic differential equation, we investigate its analytic solution by using Itô's formula and compute its numerical solution to show effects of noise. We also derive the confidence intervals for the solution of the linear scalar stochastic differential in this paper, which give more answers to why noise accelerates the convergence rate of the solution under stability with probability 1. The results in this paper suggest performance improvements of a control system in virtue of an appropriate fluctuation.