Mathematics / advanced
Probability evolving in time
Build from conditional expectation and Markov chains to martingales, Brownian motion, stochastic calculus, and simulation.
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Module 01 advanced
Probability evolving in time
Build from conditional expectation and Markov chains to martingales, Brownian motion, stochastic calculus, and simulation.
- 01
Conditional expectation as an information-dependent prediction
Define an average relative to available information, preserve its integral identities, and distinguish almost-sure equality from pointwise equality.
27 min - 02
Markov chains and stationarity
Evolve probability with a transition matrix and separate invariant distributions, recurrence, periodicity, and convergence.
28 min - 03
Poisson processes and renewals
Connect event counts to waiting times, derive the memoryless special case, and identify what average arrival rates leave unspecified.
27 min - 04
Martingales and stopping times
Express fairness relative to information, distinguish legitimate stopping rules from future selection, and apply optional stopping with its hypotheses.
30 min - 05
Brownian motion and the scale of random change
Specify a continuous Gaussian process through its increments, identify square-root time scaling, and explain why ordinary pathwise calculus fails.
29 min - 06
Itô integrals and nonanticipating accumulation
Construct integration against Brownian motion from left-endpoint decisions and use the isometry to control stochastic accumulation.
31 min - 07
Itô formula and diffusion models
Transform noisy dynamics using the quadratic-variation correction and distinguish an SDE, its solution, and a numerical approximation.
32 min - 08
Fokker–Planck equations and probability transport
Derive the forward density equation from the diffusion generator, conserve probability through flux, and calculate an Ornstein–Uhlenbeck equilibrium.
33 min - 09
Monte Carlo and variance reduction
Estimate expectations with controlled sampling error, redesign unbiased estimators, and distinguish independence, finite variance, and model error.
31 min - 10
Large deviations and rare events
Bound atypical averages with exponential moments, distinguish finite-sample inequalities from logarithmic asymptotics, and reason about rare-event simulation.
34 min