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Stochastic Calculus

Brownian motion, Ito's lemma, and SDEs.

Prerequisites: Probability + Calculus + Discrete Processes

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  1. 1Brownian MotionThe same random walk with finer and finer steps: a path that is everywhere continuous and nowhere smooth.
  2. 2Quadratic VariationAdd up the squared steps of a Brownian path and you get exactly t: the fact that breaks the ordinary chain rule.
  3. 3First Passage and the Running MaximumHow long a random walk or a stock takes to first reach a level. It always gets there but the average wait is infinite, and touching a level is twice as likely as finishing above it.
  4. 4The Itô IntegralLeft-endpoint and right-endpoint sums for ∫W dW converge to different answers, ½W_T² − ½T and ½W_T² + ½T, and finance takes the left one because a position has to be set before the price moves.
  5. 5Itô's LemmaWhy the chain rule breaks for randomness: Taylor to second order, because (dW)² = dt.
  6. 6Itô's Lemma, Part 2: Product Rule and IsometryMultiply two random processes and the product rule picks up an extra dX·dY term; square an Itô integral and the isometry turns its variance into an ordinary dt integral.
  7. 7Geometric Brownian Motion and Volatility DragUp 50% then down 40% leaves you at 90%: why the typical path of a stock grows at μ − σ²/2 while its average grows at μ.
  8. 8Radon-Nikodym on a CoinTwo coins, one tree: the ratio Z of their path probabilities turns real-world expectations into prices.
  9. 9Girsanov's TheoremTwo probability worlds, the same paths: change the measure and the drift moves, the volatility doesn’t.
  10. 10Ornstein-Uhlenbeck, Vasicek and CIRHow to solve a mean-reverting SDE with an integrating factor, and the two interest-rate models built on it: Vasicek, which can go negative, and CIR, which can't.
  11. 11Feynman-KacWhy an equation with nothing random in it computes an expected payoff: a fair game can’t drift.

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