Stochastic Integral Using One-dimensional Brownian Motion

This paper explains stochastic integral and how to calculate it using one- dimensional Brownian Motion in a partition interval matters, so we had to define a broader to category integrals on this partition interval such that it is consistent with the definition Paley Wiener and Zygmund [P-W-Z], and prove existence of this integrals by Hölder Continuous and comparison with Riemann and Stieltjes Integrals from calculus, for subsequent use in definition ITÔ'S integral which is used in the solution of most stochastic differential equations in many biological, physical fields.