| Course Description: This course will be a
applied mathematics student's exploration of notions of randomness and
random motion. Topics will include: Probability Theory: sigma algebras,
measures, construction of probability measure, dominated convergence,
Fatou's lemma, monotone convergence, condition expectation, conditional
probability, Markov chains (discrete state space), markov property,
discrete time, transition matrix, hitting times, communication classes,
irreducibility, transient and recurrent states, absorbing, positive and
null recurrence, stationary distributions, random walks, branching
processes, continuous time, Poisson process, embedded discrete time
chain, infinitesimal generator, birth & death processes, Martingale,
super and sub, Dubin's inequality, martingale convergence theorem,
Brownian motion, sample path properties, recurrence in dimension 1, 2, & 3,
stochastic differential equations, Ito calculus, Stratonovic calculus,
Hidden Markov chains/ EM , Estimation of Markov chains, Stochastic order
relations, Monte Carlo Methods, rate of convergence, error estimation,
Markov Chain Monte Carlo,simulation
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