# An Introduction to Stochastic Calculus

Through a couple of different avenues I wandered, yet again, down a rabbit hole leading to the topic of this post. The first avenue was through my main focus on a particular machine learning topic that utilized some concepts from physics, which naturally led me to stochastic calculus. The second avenue was through some projects at work in the quantitative finance space, which is one of the main applications of stochastic calculus. Naively, I thought I could write a brief post on it that would satisfy my curiosity -- that didn't work out at all! The result is this extra long post.

This post is about stochastic calculus, an extension of regular calculus to stochastic processes. It's not immediately obvious but the rigour needed to properly understand some of the key ideas requires going back to the measure theoretic definition of probability theory, so that's where I start in the background. From there I quickly move on to stochastic processes, the Wiener process, a particular flavour of stochastic calculus called Itô calculus, and finally end with a couple of applications. As usual, I try to include a mix of intuition, rigour where it helps intuition, and some simple examples. It's a deep and wide topic so I hope you enjoy my digest of it.