All Courses

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.
**Exam option: Some students may have substantial prior knowledge of engineering statistics. For them, the “Core Requirement” aspects of ORF 245, Fundamentals of Engineering Statistics, can be satisfied by a “core-requirement exam”. The 3-hour exam is administered in January to student(s) who have applied to the ORFE Director of Undergraduate Studies prior to the previous December 1 for permission to take the exam, and received authorization by the ORF 245 Exam Committee. The application should clearly state (including supporting documentation) why the student feels they already have a firm grasp of the Fundamentals of Engineering Statistics. Note: passing the exam satisfies the “Core Requirement of ORF 245” but does not contribute a course to the BSE degree’s “36 courses” requirement and it increases ORFE’s “Departmental Elective” requirement from 10 to 11, one of which must now be an advanced statistics course. In short, the “Core Requirement” for statistics can be satisfied by either taking and passing ORF 245 or for those with sufficient advanced knowledge, passing the ORF 245 exam and passing an advanced statistics course.
Many real-world problems involve maximizing a linear function subject to linear equality and/or inequality constraints. Such problems are called Linear Programming (LP) problems. Examples include min-cost network flows, portfolio optimization, options pricing, assignment problems and two-person zero-sum games to name but a few. The theory of linear programming will be developed with a special emphasis on duality theory. Attention will be devoted to efficient solution algorithms. These algorithms will be illustrated on real-world examples such as those mentioned.
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
A survey of quantitative approaches for making optimal decisions under uncertainty, including decision trees, Monte Carlo simulation, and stochastic programs. Forecasting and planning systems are integrated in the context of financial applications. Machine learning methods are linked to the stochastic optimization models.
Financial Mathematics is concerned with designing and analyzing products that improve the efficiency of markets, and create mechanisms for reducing risk. This course develops quantitative methods for these goals: the notions of arbitrage and risk-neutral pricing in discrete time, specific models such as Black-Scholes and Heston in continuous time, and calibration to market data. Credit derivatives, the term structure of interest rates, and robust techniques in the context of volatility options will be discussed, as well as lessons from the financial crisis.
The amount of data in our world has been exploding and analyzing large data sets is a central challenge in society. This course introduces the statistical principles and computational tools for analyzing big data. Topics include statistical modeling and inference, model selection and regularization, scalable computational algorithms, and more.
Predictive analytics and decision modeling are two key components of business analytics. They provide decision makers with the fundamental rationality in evaluating performance, making decisions, designing strategies, and managing risk. ORF360 focuses on both popular decision models arising from real applications, as well as mathematical decision-making tools and concepts. The first half of the course introduces the basic decision models in revenue management and pricing. The second half of the course consists of a series of business themes, such as airlines, finance, healthcare, and games, and focuses on data-driven real-world applications.
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.
Electronic commerce, traditionally the buying and selling of goods using electronic technologies, extends to essentially all facets of human interaction when extended to services, particularly information. The course focuses on both the software and the hardware aspects of traditional aspects as well as the broader aspects of the creation, dissemination and human consumption electronic services. Covered will be the physical, financial and social aspects of these technologies
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).
This is an introduction to the stochastic models inspired by the dynamics of resource sharing. Topics discussed include: early motivating communication systems (telephone and computer networks); modern applications (call centers, healthcare operations, and urban planning for smart cities); and key formulas (from Erlang blocking and delay to Little's law). We also review supporting stochastic theories like equilibrium Markov chains along with Markov, Poisson and renewal processes.
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.
Students will develop mathematical modeling skills in the context of sequential decisions under uncertainty. Students will learn the five elements of a sequential decision problem: state variables, identifying and modeling decisions, uncertainty quantification, creating transition functions, and designing objective junctions. They will learn how to design policies, and the principles of policy search and evaluation in both offline and online settings. All concepts will be taught through a series of carefully chosen problems designed to bring out specific modeling features.
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.
An introduction to the theory and practice of high frequency trading in modern electronic financial markets. We give an overview of the institutional landscape and basic empirical features of modern equity, futures, and fixed income markets. We discuss theoretical models for market making and price formation. Then we dig into detailed empirical aspects of market microstructure and how these can be used to construct effective trading strategies. Course work will be a mixture of theoretical and data-driven problems. Programming environment will be a mixture of the R statistical environment, with the Kdb database language.
This course is an introduction to commodities markets (oil, gas, metals, electricity, etc.), and quantitative approaches to capturing uncertainties in their demand and supply. We start from a financial perspective, and traditional models of commodity spot prices and forward curves. Then we cover modern topics: game theoretic models of energy production (OPEC vs. fracking vs. renewables); quantifying the risk of intermittency of solar and wind output on the reliability of the electric grid (mitigating the duck curve); financialization of commodity markets; carbon emissions markets. We also discuss economic and policy implications.
Studied is the transportation sector of the economy from a technology and policy planning perspective. The focus is on the methodologies and analytical tools that underpin policy formulation, capital and operations planning, and real-time operational decision making within the transportation industry. Case studies of innovative concepts such as "value" pricing, real-time fleet management and control, GPS-based route guidance systems, automated transit systems and autonomous vehicles will provide a practical focus for the methodologies. Class project in lieu of final exam focused on major issue in Transportation Systems Analysis.
In recent years, financial services have significantly evolved through data-driven innovation. This course will review the landscape broadly and then focus on case studies in automated lending enabled by machine learning. Students will study specific machine learning methods in current use and will be introduced to issues of fairness and explainability, which are increasingly important in this sector and beyond. Overall, the class will integrate mathematical finance models, machine learning methods and practical industry perspectives.
An introduction to the theory and practice of high frequency trading in modern electronic financial markets. We give an overview of the institutional landscape and basic empirical features of modern equity, futures, and fixed income markets. We discuss theoretical models for market making and price formation. Then we dig into detailed empirical aspects of market microstructure and how these can be used to constructe effective trading strategies. Course work will be a mixture of theoretical and data-driven problems. Programming environment will be a mixture of the R statistical environment, with the Kdb database language.
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.
Students conduct a one-semester project. Topics chosen by students with approval of the faculty. A written report is required at the end of the term.
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.
Econometric and statistical methods as applied to finance. Topics include: Asset returns and efficient markets, linear time series and dynamics of returns, volatility models, multivariate time series, efficient portolios and CAPM, multifactor pricing models, portfolio allocation and risk assessment, intertemporal equilibrium models, present value models, simulation methods for financial derivatives, econometrics of continuous time finance.
Heavy tailed distributions and copulas. Simple and multiple linear regressions. Nonlinear regression. Non-parametric regression and classification. Time series analysis: stationary and classical linear models (AR, MA, ARMA, ..). Nonlinear and non-stationary time series models. State space systems, hidden Markov models and filtering.
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.
Under the direction of a faculty member, each student carries out research and presents the results. Directed Research II has to be taken before the General Exam.
The course covers the pricing and hedging of advanced derivatives, including topics such as exotic options, greeks, interest rate and credit derivatives, as well as risk management. The course further covers basics of stochastic calculus necessary for finance. Designed for Masters students.
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.
Topics discussed include: the simplex method and its complexity, degeneracy, duality, the revised simplex method, convex analysis, game theory, network flows, primal-dual interior point methods, first order optimality conditions, Newton's method, KKT conditions, quadratic programming, and convex optimization. A broad spectrum of applications are presented.
A mathematical introduction to convex, conic, and nonlinear optimization. Topics include convex analysis, duality, theorems of alternatives and infeasibility certificates, semidefinite programming, polynomial optimization, sum of squares relaxation, robust optimization, computational complexity in numerical optimization, and convex relaxations in combinatorial optimization. Applications drawn from operations research, dynamical systems, statistics, and economics.
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.
A theoretical introduction to statistical machine learning for data science. It covers multiple regression, kernel learning, sparse regression, high dimensional statistics, sure independent screening, generalized linear models, covariance learning, factor models, principal component analysis, supervised and unsupervised learning, deep learning, and related topics such as community detection, item ranking, and matrix completion.These methods are illustrated using real world data sets and manipulation of the statistical software R.
A theoretical introduction to statistical learning problems related to regression and classification. Topics covered may include: Principle Component Analysis, nonparametric estimation, sparse regression, and Classification and Statistical learning.
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.
An introduction to stochastic calculus based on Brownian motion.Topics include:construction of Brownian motion; martingales in continuous time; the Ito integral; localization; Ito calculus; stochastic differential equations; Girsanov's theorem; martingale representation; Feynman-Kac formula.
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.
A survey of central topics in the area of investment management and financial planning. Integrating pricing methodologies with financial planning models. Linking asset and liability strategies to achieve investment goals and meet liabilities. We model the enterprise as a multi-stage stochastic program with decision strategies.
This course covers the basic concepts of modeling, measuring and managing financial risks. Topics include portfolio optimization (mean variance approach and expected utility), interest rate risk, indifference pricing, risk measures, systemic risk. Algorithms from machine learning are introduced and linked to stochastic planning models.
An introduction to analytical and computational methods common to financial math problems. Aimed at PhD students and advanced masters students who have studied stochastic calculus, the course focuses on uses of partial differential equations: their appearance in pricing financial derivatives, their connection with Markov processes, their occurrence as Hamilton-Jacobi-Bellman equations in stochastic control problems and stochastic differential games, and analytical, asymptotic, and numerical techniques for their solution.
This course discusses the formulation and the solution techniques to a wide ranging class of optimal control problems through several illustrative examples from economics and engineering, including: Linear Quadratic Regulator, Kalman Filter, Merton Utility Maximization Problem, Optimal Dividend Payments, Contact Theory. The method of dynamic programming and Pontryagin maximum principle are outlined. Brief introductions to the general theory of backward and forward stochastic differential equations, related partial differential and viscosity theory are discussed again through the above examples.
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 course provides a unified presentation of stochastic optimization, cutting across classical fields including dynamic programming (including Markov decision processes), stochastic programming, (discrete time) stochastic control, model predictive control, stochastic search, and robust/risk averse optimization, as well as related fields such as reinforcement learning and approximate dynamic programming. Also covered are both offline and online learning problems. Considerable emphasis is placed on modeling and computation.
An introduction to the microstructure of modern electronic financial markets and high frequency trading strategies. Topics include market structure and optimization techniques used by various market participants, tools for analyzing limit order books at high frequency, and stochastic dynamic optimization strategies for trading with minimal market impact at high and medium frequency. The course makes essential use of high-frequency futures data, accessed using the Kdb+ database language. Graduate credit requires completion of extended and more sophisticated homework assignments.
An introduction to nonasymptotic methods for the study of random structures in high dimension that arise in probability, statistics, computer science, and mathematics. Emphasis is on developing a common set of tools that has proved to be useful in different areas. Topics may include: concentration of measure; functional, transportation cost, martingale inequalities; isoperimetry; Markov semigroups, mixing times, random fields; hypercontractivity; thresholds and influences; Stein's method; suprema of random processes; Gaussian and Rademacher inequalities; generic chaining; entropy and combinatorial dimensions; selected applications.
Markov processes with general state spaces; transition semigroups, generators, resolvants; hitting times, jumps, and Levy systems; additive functionals and random time changes; killing and creation of Markovian motions.
Recent developments in the theory and applications of the analysis of random processes and random fields. Applications include financial engineering, transport by stochastic flows, and statistical imaging.
Empirical Process theory mainly extends the law of large numbers (LLN), central limit theorem (CLT) and exponential inequalities to uniform LLN's and CLT's and concentration inequalities. This uniformaty is useful to statisticians and computer scientists in that they often model data as a sample from some unknown distribution and desire to estimate certain aspects of the population. Uniform LLN or CLT and concentration inequalities will imply that certain sample averages will be uniformly close to their expectations regardless of the unknown distributions. This class intends to review modern empirical process theory and its related asymptot
Course is on statistical theory and methods for high-dimensional statistical learning and inferences arising from processing massive data from various scientific disciplines. Emphasis is given to penalized likelihood methods, independence screening, large covariance modeling, and large-scale hypothesis testing. The important theoretical results are proved.
Starting from the classical mathematical theory of games, the first part of the course covers auction theory and focus on games with a large number of agents. The second part of the course concentrates on dynamic games (starting from Markov Decision Processes and Reinforcement Learning) and end with Mean Field Control problems and Mean Field Games.
The seminar course presents recent developments in the theory of large stochastic systems. The first part of the course is focused on the equilibrium theory of game models when the interactions between the agents is through a large network modeled with a weighted random graph. The second part of the course focuses on probabilistic models with singular mean field interactions and applications to free boundary problems in material science, financial networks and neuroscience.
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.
Algorithms for approximate dynamic programming/reinforcement learning come in a number of styles. This seminar will survey the literature on the theory supporting the convergence of different algorithms on different classes of problems drawing from computer science, engineering, economics and operations research. Each week will consist of a presentation by one of the students that offers a convergence proof for a particular algorithm/problem class.
This is a graduate course focused on research in statistical optimization and reinforcement learning, particularly in a large-scale setting. We discuss both theoretical and algorithmic tools to address these problems. Specific topics include: (1) randomized linear algebra (2) spectral method (3) tensor decomposition and mixture models (4) distributed estimation and optimization (5) complexity of Markov decision process (6) imitation learning (7) graph sparsification theory. Students are required to participate in paper surveying and presentation.
The course explores the modelling, measuring and managing of financial risk - i.e., differentiating between a good trade and a bad one. Emphasis is placed on illustrating the modelling needs and strategies on the buy side vs. the sell side. Some elements of Machine Learning and its application to financial time series is discussed. Practical applications are drawn from all asset classes. Convexity, entropy and time scale considerations are addressed with respect to RV and macro strategies. Coursework focuses on assessing risk/reward and identifying market opportunities. Guest speakers from hedge funds and banks.