Continuous Time Finance

This is a graduate course on the mathematical machinery that makes modern derivative pricing possible: how to do calculus when the thing you're integrating against is as wild as Brownian motion. You build up from discrete martingales and stopping times, work through the construction of the Itô integral and its formula, and end by deriving the Black-Scholes model as a payoff of all that scaffolding. Assessment is six problem sets plus a written final, and the work is heavily proof-driven, expect to spend real time on convergence theorems and martingale arguments before any "finance" shows up. It assumes solid measure-theoretic probability and feeds directly into research in mathematical finance, stochastic control, and anything where continuous-time noise matters.