Approximative Theorem of Incomplete Riemann-Stieltjes Sum of Stochastic Integral

TL;DR

This paper investigates the convergence of incomplete Riemann-Stieltjes sums to stochastic integrals, providing sufficient conditions and demonstrating with simulations for Ito and Stratonovich integrals.

Contribution

It develops new sufficient conditions for incomplete sums to approximate stochastic integrals, offering alternative convergence methods.

Findings

Established conditions for convergence of incomplete sums to stochastic integrals.

Provided simulation examples demonstrating the approximation methods.

Extended understanding of stochastic integral approximation techniques.

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.