An introduction to the modern theory of asset pricing. Topics include: No arbitrage, Arrow-Debreu prices and equivalent martingale measure; security structure and market completeness; mean-variance analysis, Beta-Pricing, CAPM; and introduction to derivative pricing.

A first introduction to probability and statistics. This course will provide background to understand and produce rigorous statistical analysis including estimation, confidence intervals, hypothesis testing and regression and classification. Applicability and limitations of these methods will be illustrated using a variety of modern real world data sets and manipulation of the statistical software R.

An introduction to probability and its applications. Topics include: basic principles of probability; Lifetimes and reliability, Poisson processes; random walks; Brownian motion; branching processes; Markov chains

An introduction to several fundamental and practically-relevant areas of modern optimization and numerical computing. Topics include computational linear algebra, first and second order descent methods, convex sets and functions, basics of linear and semidefinite programming, optimization for statistical regression and classification, and techniques for dealing with uncertainty and intractability in optimization problems. Extensive hands-on experience with high-level optimization software. Applications drawn from operations research, statistics and machine learning, economics, control theory, and engineering.

Independent research or investigation resulting in a substantial formal report in the student's area of interest under the supervision of a faculty member.

This course showcases how networks are widespread in society, technology, and nature, via a mix of theory and applications. It demonstrates the importance of understanding network effects when making decisions in an increasingly connected world. Topics include an introduction to graph theory, game theory, social networks, information networks, strategic interactions on networks, network models, network dynamics, information diffusion, and more.

An introduction to popular statistical approaches in regression and time series analysis. Topics will include theoretical aspects and practical considerations of linear, nonlinear, and nonparametric modeling (kernels, neural networks, and decision trees).

An introduction to the uses of simulation and computation for analyzing stochastic models and interpreting real phenomena. Topics covered include generating discrete and continuous random variables, stochastic ordering, the statistical analysis of simulated data, variance reduction techniques, statistical validation techniques, nonstationary Markov chains, and Markov chain Monte Carlo methods. Applications are drawn from problems in finance, manufacturing, and communication networks. Students will be encouraged to program in Python. Office hours will be offered for students unfamiliar with the language.

This course develops several methods that are central to modern optimization and learning problems under uncertainty. These include dynamic programming, linear quadratic regulator, Kalman filter, multi-armed bandits and reinforcement learning. Representative applications and numerical methods are emphasized.

This course covers the basic concepts of measuring, modeling and managing risks within a financial optimization framework. Topics include single and multi-stage financial planning systems. Implementation from several domains within asset management and goal based investing. Machine learning algorithms are introduced and linked to the stochastic planning models. Python and optimization exercises required.

Over recent decades, novel data sources and machine learning models have enabled rapid evolution across financial services. This course focuses on ongoing innovations in consumer lending. The technical material spans finance and machine learning topics, including fairness and explainability, important in lending and also more broadly. The class integrates technical topics with critical explorations of business practices informed by readings, class discussion and outside speakers, and includes work with industry data sets.

A formal report on research involving analysis, synthesis, and design, directed toward improved understanding and resolution of a significant problem. The research is conducted under the supervision of a faculty member, and the thesis is defended by the student at a public examination before a faculty committee. The senior thesis is equivalent to a year-long study and is recorded as a double course in the Spring.

The course is divided into three parts of approximately the same lengths. Density estimation (heavy tail distributions) and dependence (correlation and copulas). Regression analysis (linear and robust alternatives, nonlinear, nonparametric,classification.) Machine learning (TensorFlow, neural networks, convolution networks and deep learning). The statistical analyzes, computations and numerical simulations are done in R or Python.

Under the direction of a faculty member, each student carries out research and presents the results. Directed Research is normally taken during the first year of study.

This course introduces analytical and computational tools for linear and nonlinear optimization. Topics include linear optimization modeling, duality, the simplex method, degeneracy, sensitivity analysis and interior point methods. Nonlinear optimality conditions, KKT conditions, first order and Newton's methods for nonlinear optimization, real-time optimization and data-driven algorithms. A broad spectrum of applications in engineering, finance and statistics is presented.

A graduate-level introduction to statistical theory and methods and some of the most important and commonly-used principles of statistical inference. Covers the statistical theory and methods for point estimation, confidence intervals (including modern bootstrapping), and hypothesis testing. These topics will be covered in both nonparametric and parametric settings, and from asymptotic and non-asymtoptotic viewpoints. Basic ideas from measure-concentration and notions of capacity of functional classes (e.g. VC, covering and bracketting numbers) will be covered as needed to support the theory.

This is a graduate introduction to probability theory with a focus on stochastic processes. Topics include: an introduction to mathematical probability theory, law of large numbers, central limit theorem, conditioning, filtrations and stopping times, Markov processes and martingales in discrete and continuous time, Poisson processes, and Brownian motion.

The intent of this course is to introduce the student to the technical and algorithmic aspects of a wide spectrum of computer applications currently used in the financial industry, and to prepare the student for the development of new applications. The student is introduced to C++, the weekly homework involves writing C++ code, and the final project also involves programming in the same environment.

This course covers the basic concepts of measuring, modeling and managing risks within a financial optimization framework. Topics include single and multi-stage financial planning systems. Implementation from several domains within asset management and goal based investing. Machine learning algorithms are introduced and linked to the stochastic planning models. Python and optimization exercises required.

This course is an introduction to deep learning theory. Using tools from mathematics (e.g. probability, functional analysis, spectral asymptotics and combinatorics) as well as physics (e.g. effective field theory, the 1/n expansion, and the renormalization group) we cover topics in approximation theory, optimization, and generalization.

This introductory course on the theory of deep learning, emphasizes the integrated nature of theory, exploratory empirical work, and concrete solutions to difficult machine learning problems.