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Stochastic Processes
Explore processes and dependence; counting and continuous paths; information and stochastic calculus. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Random variables
A random variable is a measurable function on a probability space, allowing outcomes to be described numerically.
5 reviewed resources
Processes and dependence
Stochastic processes
A stochastic process is a time-indexed collection of random variables.
Video explanations to be addedMarkov chains
A Markov chain's next-state distribution depends on the present state rather than the full history.
Video explanations to be addedTransition matrices
A finite Markov chain uses a nonnegative matrix with row sums one to describe transitions.
Video explanations to be addedStationary distributions
A stationary distribution remains unchanged after applying the chain's transition rule.
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Counting and continuous paths
Poisson processes
A homogeneous Poisson process counts events with stationary independent increments at a constant rate.
Video explanations to be addedBrownian motion
Brownian motion has continuous paths and independent Gaussian increments with variance equal to elapsed time.
Video explanations to be addedRenewal processes
Independent identically distributed waiting times generate repeated renewal events.
Video explanations to be addedQueueing processes
Arrival and service mechanisms determine random queue lengths and waiting times.
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Information and stochastic calculus
Filtrations
An increasing family of sigma-algebras represents information available over time.
Video explanations to be addedMartingales
An integrable adapted process has conditional expected future value equal to its present value.
Video explanations to be addedStopping times
Whether a stopping time has occurred by a time is determined by information available then.
Video explanations to be addedItô calculus
Stochastic integration against Brownian motion includes a quadratic-variation correction in change-of-variable formulas.
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