Stochastic Integrals
Introduction to Stochastic Calculus
Introduction
Now that we know how to define Martingales and Markov Processes, we can begin to define Brownian motion. A random walk model is a Markov process, and without a drift they are Martingales. In finance, the random walk hypothesis states stock prices evolve according to a random walk model without a drift so changes in stock prices cannot be predicted. Brownian motion is a random walk in continuous time, each sample path is continuous and the net change over any future period is unpredictable. These models and the accompanying theories are the core of quantitative finance. Essentially every pricing model and piece of advanced coursework revolves around assumptions made by these models. Since this is the case, in this article, I will introduce the idea of Brownian motion and a stochastic integral along with their purpose. Please note that this math, even for those who have taken advanced mathematics coursework into graduate school, is very difficult. Do your best to follow along, and it will infinitely help to write out the math as you go.
Brownian Motion
Brownian Motion Conditions
The definition of Brownian motion follows these assumptions…
- Consider indexes of time s,t such that…
