# Probability Classes At Duke

• MATH 230/STA 230: Probability
• MATH 340: Advanced Introduction to Probability
• MATH 541: Applied Stochastic Processes
• MATH 545: Stochastic Calculus
• MATH 581: Mathematical Finance
• STAT 711: Probability & Measure Theory
• MATH 641: Probability
• STAT 961: Statistical Stochastic Processes
• MATH 690-40: Topics in probability

## Classes typically taken by Undergraduates

MATH 230/STA 230 Typical first course in probability. This is the entry course for a statistics major or minor as MATH 342/STA 250 requires MATH/STA 230
MATH 340 A replacement for Math 230 which is more theoretical and covers topics in more depth,. It is also fulfills the prerequisites for MATH 342/STA 250.
MATH 541 Building on Math 230/340, gives an introduction to stochastic processes. The typical class which follows the introductory probability class. Stochastic process are a basic tool of probabilistic modeling and analysis. They are used in Statistics, Ecology, Genetics, Management Science, Economics, Computer Science and Electrical engineering to name a few disciplines of applicability.
MATH 545 Gives a useful, application oriented introduction to the theory of stochastic calculus. Stochastic calculus is used in population biology, finance, econometric,and physics to name a few disciplines of applicability. Requires more analysis than Math 541, but is independent of that class. If both Math 541 and Math 545 are to be taken, it is preferable to take Math 541 first, thought it is not required.
MATH 581 Despite its higher number, this class assumes less than Math 541 or Math 545. It concentrates on the stochastic process used in mathematical finance.

## Classes typically taken by Masters students and Phd Students

MATH 541 Basic masters level stochastic process class. Targeted at PhD and masters students from around the university. Requires and introduction to probability. Not intended for math PhD students intending to continue doing research in probability. Perfect for mathematics PhD students who want a background in stochastic modeling to compliment other interests.
MATH 545 Targeted at both masters students and Phd students in mathematics, statistics and other departments. Suitable for students conducting research in probability as well as those who are working in more applied domains. Requires basic probability and some exposure to PDEs and mathematical analysis. Graduate real analysis is not required but is useful.
MATH 581 Despite its higher number, this class assumes less than Math 541 or Math 545. It concentrates on the stochastic process uses in mathematical finance. It is directed at masters and Phd students in other departments and is suitable for mathematics PhD students who want to learn more about finance. It is not part of the Graduate level sequence in probability for Mathematics PhD students intending to work in probability but can provide a background in mathematical finance for them.
MATH 531 Though not a probability course, Basic Analysis I does contain about 2-3 weeks of probability at the end. It is a more complete introduction to graduate real analysis and measure theory than STATS 711 but does not cover near as much introductory probability. This can be rectified by watching the supplementary videos which can be found here.
STATS 711 This is a graduate introduction to basic measure theory and graduate measure theoretic probability. It is a typical first class for those interested in getting a PhD level introduction to measure theoretic probability.
MATH 641 This is a second course in measure theoretical probability. It is meant to follow STATS 711 but it is taught so that it can be taken by people who did not take 711. The topics it covers includes Martingales, Markov Processes, Brownian Motion, 0-1 laws, branching processes, and ergodic theory. It is also open to students who have obtained a sufficient introduction to measure theoretical probability my other means.
STATS 961 This is a course in stochastic processes assuming a grasp of measure theoretic probability. It concentrates the theory and modeling aspects of Stochastic Processes central to Bayesian statistical analysis. The topics shift from year to year and is generally disjoint from the topics in MATH 641. Examples of possible topics are:
• Convergence theory of Markov chains in general state spaces;
• Convergence rates and mixing times for Markov chains;
• Stochastic approximation and adaptive estimators;
• Inference and prediction for Gaussian Processes and Random Fields;
• Modeling and Inference with Continuous-time Jump Processes;
• Theory, Application, and inference for Levy Random Flights;
• Inference for Diffusions;
• Exotic time series with Infinitely Divisible innovations;
• Functional Data Analysis.
MATH 690-40 This is a topics course which is offered every Fall. The topics change from year to year. Some years it is appropriate for those with a non-measure theoretic introduction to probability. Other years it assume more knowledge. Previous topics include: Random Graphs, Stochastic PDEs and Erogidic Theory, Data and Random Matrices, Particle systems, and Random Dynamical systems.

## Possible Paths through the PhD probability classes

Option 1:
1. Fall (1st Year) - STAT 711 : Probability & Measure Theory
2. Fall (1st Year) - Math 631 Real : Analysis
3. Spring (1st Year)- MATH 641 (287): Probability
Option 2:
1. Fall (1st Year) - STAT 711 : Probability & Measure Theory
2. Spring (1st Year)- MATH 641 (287): Probability
Option 3:
1. Fall (1st Year) - Math 631 Real : Analysis
2. Watch Supplemental (Winter break)- Video and reading
3. Spring (1st Year) - MATH 641 (287): Probability

Option 1 is for a student who wants a solid back ground in graduate probability and real analysis.

Option 2 is for a student who is manly interested in probability and not getting a PhD in mathematics but likely taking upper level clases in Stats.

Option 3 is for a student who is confident of there back ground and wants a solid foundation in probability and real analysis.

Stochastic Calculus (MATH 545) is a masters level class which is still very appropriate for PhD students. It may be taken at any time assuming the one has undergraduate/master level probability and real analysis..

## Possible Paths through the Master probability classes

The undergraduate classes Math 230, Math 340, and Math 431 are often taken by less mathematically prepared students to improve their background.

Applied Stochastic Processes (MATH 541) is mainly targeted at masters and PhD students from outside of mathematics and mathematics PhD students who are not concentrating in graduate probability but what the background needed for basic stochastic modeling.

Stochastic Calculus (MATH 545) is a masters level class which require a little more sophistication than MATH 541 in that it assumes some undergraduate analysis. It can be taking with or with out MATH 541. If both are to be taken it is most reasonable to take Math 545 second, though not required.