Unit 5 of 14

Stochastic calculus

  • 8 lessons
  • Videos 33 to 40

Continuous-time models need precise rules for information, noise and integration. This unit builds Brownian motion, Itô integrals, Itô's lemma, stochastic differential equations and measure changes.

Filtrations, predictable processes and stopping times separate decisions on current information from hindsight. Brownian motion has independent Gaussian increments and a nonzero quadratic variation, which produces the extra term in stochastic calculus. The Itô integral is built from simple predictable integrands through the Itô isometry, and Itô's lemma adds a second-order correction to the chain rule. Stochastic differential equations have strong and weak solutions. Girsanov's theorem changes the drift under conditions such as Novikov's. General semimartingale theory is beyond the course.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Information, Brownian motion and quadratic variation

  1. Stochastic processes filtrations and stopping times

    How a filtration records information, when a process is adapted or predictable, what the predictable sigma algebra and the usual conditions are, which random times are stopping times, and why anticipatory strategies fail.

    Video 33

  2. Brownian motion and Gaussian increments

    What a standard Brownian motion is, how its independent Gaussian increments give the transition density, the covariance and square-root-of-time scaling, and how to compute a conditional threshold probability.

    Video 34

  3. Quadratic variation and stochastic limits

    How sums of squared Brownian increments converge in mean square and in probability to elapsed time as the partition mesh shrinks, why finite variation paths give zero, and how quadratic covariation works for diffusions.

    Video 35

Itô integrals and stochastic differential equations

  1. The Ito integral and isometry

    How the Itô integral is built from simple predictable integrands and mean-square limits, why the Itô isometry and the martingale property hold for square-integrable integrands, and why the left endpoint matters.

    Video 36

  2. Ito lemma and the stochastic chain rule

    How Itô's lemma extends the chain rule to Itô processes, where its second-order correction comes from, and how the multivariate Itô formula, the Itô product rule and the generator of a diffusion follow from it.

    Video 37

  3. Stochastic differential equations and solution concepts

    What it means to solve a stochastic differential equation, strong and weak solutions, the global Lipschitz theorem on existence and pathwise uniqueness, the Ornstein-Uhlenbeck process and the Euler step.

    Video 38

Measure changes and Girsanov's theorem

  1. Exponential martingales and measure changes

    How a Radon-Nikodym derivative reweights the same events, the conditional Bayes formula, the stochastic exponential, when it is a true exponential martingale, and the Novikov condition.

    Video 39

  2. Girsanov theorem and risk neutral dynamics

    Girsanov's theorem with its density hypotheses and sign convention, how a measure change shifts a diffusion drift, the market price of risk, risk-neutral stock dynamics, measure equivalence and Lévy's characterization.

    Video 40