A Capstone Project

The goal of the capstone project

The goal of the capstone project for our group is to investigate the Riemann integral and Itô's integral. Using what was learned in real analysis class, a Riemann integral approximates the area under the graph of the function between a and b by summing the areas of all the rectangles. This method is similar to calculus and includes dividing an interval a to b into smaller intervals. The Itô's integral uses either left-hand or right-hand approaches to approximate the area under the graph in a manner similar to that of the Itô's integral. The Riemann and Itô's integrals have some discrepancies despite their similarity. While Riemann integral can work with a function without the random part, Itô's integral can only work with a function without the random part, Itô's integral can only work with a function in the presence of some random processes. We, therefore, intend to divide our project into three parts.

Review of Riemann integral

First, we will review the construction of Riemann integral and some of its fundamental properties, such as Riemann sum, Riemann integrability, as well as the Fundamental Theorem of Calculus. We will then solve some problems relating to Riemann integral as examples, and prove some of the Riemann integral's definitions.

Background information about probability theory and Brownian motion

Second, we will discuss and give some background information about probability theory and Brownian motion. We will outline various properties of Brownian motion and provide some results of Brownian motion without proof. We will then discuss the construction of Itô's integral, as well as outline its fundamental properties. Besides, we will explain stochastic differential equations to give a better understanding of Itô's integral.

Proving Itô's formula and discussing applications

Finally, we will prove Itô's formula, which is essential in computing Itô's integral. We will then discuss some applications of Itô's integral and stochastic differential equations, including how Itô's integral is used in the stock market, as well as its application to Optimal Stopping, Black-Scholes model, or Kalwan-Filtering.