01
Stochastic Processes in Continuous Time
Foundations: paths, filtrations, martingales, stopping times, and Brownian motion.
6 sections
§ 01 / Learn
The curriculum.
Structured notes from continuous-time foundations through Feynman–Kac. Progress is saved locally in your browser.
02
Martingales in Continuous Time
Optional sampling, Doob inequalities, convergence, and applications to Brownian motion.
8 sections
03
Stochastic Integration
Why pathwise integrals fail, the Itô isometry, and the continuous martingale integral.
5 sections
04
Localization and Local Martingales
Extending the Itô integral beyond L² and the language of local martingales.
4 sections
05
The Itô Formula
Change of variables for stochastic processes, multivariate forms, and quadratic variation.
7 sections
06
Stochastic Differential Equations
GBM, Ornstein–Uhlenbeck, Brownian bridge, existence/uniqueness, and numerical schemes.
11 sections
07
Girsanov's Theorem
Change of measure, exponential tilting, Novikov's condition, and drifted Brownian motion.
13 sections
08
Martingale Representation & Doob–Meyer
Representation of martingales, Lévy's characterization, and predictable quadratic variation.
7 sections
09
The Diffusion Equation
Heat equation derivation, solutions via Gaussians, uniqueness, and maximum principles.
6 sections
10
Feynman–Kac & the Dirichlet Problem
Probabilistic representations of PDEs, arcsine law, harmonic functions, and Dirichlet problems.
8 sections