Approximative Theorem of Incomplete Riemann-Stieltjes Sum of Stochastic Integral

Abstract: The approximative theorems of incomplete Riemann-Stieltjes sums of Ito stochastic integral, mean square integral and Stratonovich stochastic integral with respect to Brownian motion are investigated. Some sufficient conditions of incomplete Riemann-Stieltjes sums approaching stochastic integral are developed, which establish the alternative ways to converge stochastic integral. And, Two simulation examples of incomplete Riemann-Stieltjes sums about Ito stochastic integral and Stratonovich stochastic integral are given for demonstration.