§ 00 — A Visual Field Guide

Stochas·tic

continuous-time probability — rendered as processes, martingales, and the calculus of noise.

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From Brownian paths to Feynman–Kac.

This platform turns Keller-Ressel's Stochastic Analysis notes into an interactive field guide: structured lessons with KaTeX, canvas simulations, quizzes with immediate feedback, practice problems, and an open discussion board — all without creating an account.

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Ten foundations.

  1. 01Stochastic Processes in Continuous TimeFoundations: paths, filtrations, martingales, stopping times, and Brownian motion.
  2. 02Martingales in Continuous TimeOptional sampling, Doob inequalities, convergence, and applications to Brownian motion.
  3. 03Stochastic IntegrationWhy pathwise integrals fail, the Itô isometry, and the continuous martingale integral.
  4. 04Localization and Local MartingalesExtending the Itô integral beyond L² and the language of local martingales.
  5. 05The Itô FormulaChange of variables for stochastic processes, multivariate forms, and quadratic variation.
  6. 06Stochastic Differential EquationsGBM, Ornstein–Uhlenbeck, Brownian bridge, existence/uniqueness, and numerical schemes.
  7. 07Girsanov's TheoremChange of measure, exponential tilting, Novikov's condition, and drifted Brownian motion.
  8. 08Martingale Representation & Doob–MeyerRepresentation of martingales, Lévy's characterization, and predictable quadratic variation.
  9. 09The Diffusion EquationHeat equation derivation, solutions via Gaussians, uniqueness, and maximum principles.
  10. 10Feynman–Kac & the Dirichlet ProblemProbabilistic representations of PDEs, arcsine law, harmonic functions, and Dirichlet problems.