Course Descriptions - Probability Theory/Probabilistic Models
550.420 Introduction to Probability: Probability and its applications, at the calculus level. Emphasis on techniques of application rather than on rigorous mathematical demonstration. Probability, combinatorial probability, random variables, distribution functions, important probability distributions, independence, conditional probability, moments, covariance and correlation, limit theorems. Students initiating graduate work in probability or statistics should enroll in 550.620. Prerequisite: one year of calculus. Recommended corequisite: multivariable calculus.
550.426 Introduction to Stochastic Processes: Mathematical theory of stochastic processes. Emphasis on deriving the dependence relations, statistical properties, and sample path behavior including random walks, Markov chains (both discrete and continuous time), Poisson processes, martingales, and Brownian motion. Applications that illuminate the theory. Prerequisite: 550.420.
550.427 Stochastic Processes and Applications to Finance: A development of stochastic processes with substantial emphasis on the processes,concepts, and methods, useful in mathematical finance. Relevant concepts from probability theory, particularly conditional probability and conditional expectation, will be briefly reviewed. Important concepts in stochastic processes will be introduced in the simpler setting of discrete-time processes, including random walks, Markov chains, and discrete-time martingales, and these will be used to motivate more advanced material. Most of the course will concentrate on continuous-time stochastic processes, particularly martingales, Brownian motion, diffusions, and basic tools of stochastic calculus. Examples will focus on applications in finance, economics, business, and actuarial science. Prerequisite: 550.420 Introduction to Probability
550.440 Stochastic Calculus: Introduction to stochastic integration, stochastic differential equations, and the Ito calculus. Emphasis will be on underlying ideas rather than rigorous development. Stochastic processes, Brownian motion, conditional expectation, martingales, Ito and Stratonovich integrals and their calculus, stochastic differential equations. Some applications to finance, stochastic flow systems, or other areas should be provided. Prerequisites: 550.420; stochastic processes recommended, but not required.
550.620 Probability Theory I: Probability as a mathematical discipline, including introductory measure theory. Axiomatic probability, combinatorial probability, random variables, conditional probability, independence, distribution theory, expectation, Lebesgue-Stieltjes integration, variance and moments, probability inequalities, characteristic functions, conditional expectation. Prerequisites: 110.405 and 550.420, or equivalents.
550.621 Probability Theory II: Probability at the level of measure theory, focusing on limit theory. Modes of convergence, Poisson convergence, three-series theorem, strong law of large numbers, continuity theorem, central limit theory, Berry-Esseen theorem, infinitely divisible and stable laws. Prerequisites: 550.620, 110.405, or equivalents.
550.626 Stochastic Processes II: Measure-theoretic
treatment of stochastic processes,
focusing on continuous parameter processes. Brownian motion and diffusion,
renewal processes, continuous-time martingales, stationary processes.
Existence and construction, structure, sample path behavior, and asymptotic
properties. Prerequisites: 550.426, 550.621.
550.723 Markov Chains: Recent advances in computer science, physics, and statistics have been made possible by corresponding sharply quantitative developments in the mathematical theory of Markov chains. Possible topics: rates of convergence to stationarity, eigenvalue techniques, Markov chain Monte Carlo, perfect simulation, self-organizing data structures, approximate counting and other applications to computer science, reversible chains, interacting particle systems.