Introduction to Stochastic Calculus

This lecture introduces stochastic processes as time-indexed collections of random variables and explains how information evolves through filtrations. It covers adapted processes, martingales as “fair games,” and why conditioning plays a central role over time. The lecture then builds intuition for Brownian motion, including its defining properties, connection to random walks, and why its variance grows linearly in time. These ideas set the conceptual groundwork for stochastic integrals, Itô calculus, and stochastic differential equations.

This lecture introduces stochastic integrals and explains why ordinary calculus does not work for Brownian motion. It builds the Itô integral step by step, starting from simple adapted processes and extending to general integrands. You’ll see how non-anticipation is enforced, why left endpoints matter, and how the Itô isometry provides a clean variance identity that makes everything well-defined. This result is the main technical tool behind stochastic calculus and sets up Itô’s lemma and stochastic differential equations.

This lecture develops Itô’s lemma for time-dependent functions in one dimension and shows how it acts as the correct chain rule for stochastic processes. It explains where the extra second-derivative term comes from and how to use Itô’s formula in practice. Several reusable templates are introduced, including squaring Brownian motion, exponential martingales, drift-removal tricks, and log transforms for multiplicative noise.

This lecture explains quadratic variation and why it is the key reason stochastic calculus differs from ordinary calculus. It shows how Brownian motion accumulates variance over time, leading to rules like (dW)2=dt(dW)^2 = dt(dW)2=dt. Using this idea, the lecture connects quadratic variation to Itô’s formula and then applies these tools to solve common stochastic differential equations, including additive noise, geometric Brownian motion, and linear SDEs, using clear, reusable solution templates.

This lecture applies stochastic calculus to finance through the Black–Scholes model. It shows how modeling stock prices with geometric Brownian motion, applying Itô’s lemma, and enforcing no-arbitrage lead directly to the Black–Scholes PDE. You’ll see how delta hedging removes randomness, why the option price depends on volatility and interest rates but not the stock’s expected return, and how the pricing formula follows from a clean, reusable SDE-based argument rather than memorization.

This lecture introduces mean reversion through the Ornstein–Uhlenbeck process, a core model for systems that fluctuate randomly but are pulled back toward a baseline. It walks through the OU SDE, explains how it differs from Brownian motion, and solves it explicitly using an integrating factor. You’ll see how the mean returns to a long-run level, why the variance stabilizes instead of growing, and how this model naturally appears in neuroscience, physics, and engineering settings where noise and feedback coexist.