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MathApplied mathematics and statistics

Stochastic processes

Random walks, Markov chains, Brownian motion, quadratic variation and martingales.

3 lessons open · 20 planned

Steps 9, 10 and 11 are part of the option pricing path, a 22-step course from discounting to Black–Scholes.

Pages you can open

From random walk to Brownian motion How fast must the steps of a random walk shrink as its time step shrinks? Option pricing path, step 9 · ~40 min Martingales What makes a random process a fair game? Option pricing path, step 11 · ~45 min Quadratic variation and (dW)² = dt What is left of the squared increments of a Brownian path when the steps get very small? Option pricing path, step 10 · ~45 min

Planned

In learning order. These are not pages yet.

  1. Markov chains: watching probability flow between statesIntro
  2. Stationary distributions and the eigenvalue-1 eigenvectorAdvanced undergrad
  3. Simple random walk: where you end up after n stepsIntro
  4. Mixing time: how fast a Markov chain forgets where it startedGraduate
  5. Gambler's ruin: the chance of hitting the top firstAdvanced undergrad
  6. The reflection principle: counting paths that touch a barrierAdvanced undergrad
  7. The Poisson process: arrivals with no memoryIntro
  8. Recurrence and transience: does a random walk come home?Advanced undergrad
  9. Galton-Watson branching: when a family line dies outAdvanced undergrad
  10. Compound Poisson processes and jumpsAdvanced undergrad
  11. Continuous-time Markov chains and the generator matrixAdvanced undergrad
  12. Brownian paths: continuous everywhere, smooth nowhereAdvanced undergrad
  13. Optional stopping: why choosing when to quit can't beat a fair gameAdvanced undergrad
  14. Correlated Brownian motionsGraduate
  15. Brownian motion and the heat equation: where ½ ∂²/∂x² comes fromGraduate
  16. Martingale convergence: why a bounded fair game must settle downGraduate
  17. Gaussian processes: a random function described by its mean and covarianceGraduate
  18. First passage times of Brownian motionAdvanced undergrad
  19. The Brownian bridge: Brownian motion pinned at both endsGraduate
  20. The Ornstein-Uhlenbeck process: a random walk on a springAdvanced undergrad

How it connects

Builds on Probability Linear algebra Differential equations Stochastic calculus and SDEs Proof, logic and sets connection Real analysis connection

Leads to Stochastic calculus and SDEs Derivatives Time series PDE methods Volatility and jump models Interest-rate models Differential equations Computational statistics Arbitrage and risk-neutral pricing Monte Carlo and numerical pricing Credit risk

Subjects marked “connection” are linked by a related idea, not by a prerequisite.

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