Introduction to Stochastic Calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It is famously used in quantitative finance to model the random behavior of asset prices over continuous time.

Core Concepts

1. Brownian Motion (Wiener Process): A continuous-time stochastic process that serves as a building block for more complex models. It has independent, normally distributed increments.
2. Itô Calculus: An extension of traditional calculus to stochastic processes. Because paths of Brownian motion are not differentiable in the ordinary sense, Itô developed a new integral to handle them.
3. Itô's Lemma: The stochastic counterpart to the chain rule in ordinary calculus. It allows us to find the differential of a function of a stochastic process.
4. Stochastic Differential Equations (SDEs): Equations in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process.
5. Black-Scholes Model: The most famous application of stochastic calculus in finance, it models the price of an option over time using geometric Brownian motion.