# 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. Applicability and limitations of these methods will be illustrated using a variety of modern real world data sets and manipulation of the statistical software such as Matlab or R. Precepts are based on real data analysi.

Many real-world problems involve maximizing a linear function subject to linear 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-sumgames 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 with a focus on financial applications.

Financial Mathematics is concerned with designing and analyzing products that improve the efficiency of markets and create mechanisms for reducing risk. This course introduces the basics of quantitative finance: the notions of arbitrage and risk-neutral probability measure are developed in the case of discrete models. Black-Scholes theory is introduced in continuous-time models, and credit derivatives and the term structure of interest rates are 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 becoming a central problem in our society. This course introduces the statistical principles and computational tools for analyzing big data: the process of exploring and predicting large datasets to find hidden patterns and gain deeper understanding, and of communicating the obtained results for maximal impact. Topics include massively parallel data management and data processing, model selection and regularization, statistical modeling and inference, scalable computational algorithms, descriptive and predictive analysis, and exploratory analysis.

This is an introductory course to decision methods and modeling in business and operations management. The course will emphasize both mathematical decision-making techniques, as well as popular data-based decision models arising from real applications. Upon completion of this course students will have learned analytical tools for modeling and optimizing business decisions. From a practical perspective, this will be a first course that gives an overview of advanced operations research topics including revenue management, supply chain management, network management, and pricing.

An introduction to several fundamental and practically-relevant areas of numerical computing with an emphasis on the role of modern optimization. Topics include computational linear algebra, descent methods, basics of linear and semidefinite programming, optimization for statistical regression and classification, trajectory optimization for dynamical systems, 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, finance, 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.

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

Regression: linear, nonlinear, nonparametric. Quantile regression. Time series: classical linear models, univariate and multivariate; elements of spectral analysis; stochastic volatility models (ARCH, GARCH, ....); dynamic factor models

An introduction to the fundamental results of queueing theory. Topics covered include the classical traffic, offered load, loss and delay stochastic models for communication systems. Through concrete examples and motivations we discuss the theory of Markov chains, Poisson processes and Monte-Carlo simulation. Fundamental queueing results such as the Erlang blocking and delay formulae, Little's law and Lindley's equation are presented. Applications are drawn from communication network systems, inventory management, and optimal staffing.

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.

The management of complex systems through the control of physical, financial and informational resources. The course focuses on developing mathematical models for resource allocation, with an emphasis on capturing the role of information in decisions. The course seeks to integrate skills in statistics, stochastics and optimization using applications drawn from problems in dynamic resource management which tests modeling skills and teamwork.

Optimal learning addresses the challenge of collecting information efficiently when information is expensive. Applications include topics such as finding the best price for a product, identifying the best treatment for a disease, tuning the parameters in a bidding policy, or choosing the best player for a sports team. Students learn how to formulate a learning problem, identify a belief model, and quantify the value of information. The course covers online and offline learning problems, and introduces students to a range of policies for collecting information.

This course is an introduction to commodities markets (energy, metals, agricultural products) and issues related to renewable energy sources such as solar and wind power, and carbon emissions. Energy and other commodities represent an increasingly important asset class, in addition to significantly impacting the economy and policy decisions. Emphasis will be on the application of Financial Mathematics to a variety of different products and markets. Topics include: energy prices (including oil and electricity); cap and trade markets; storable vs non-storable commodities; financialization of commodities markets; applications of game theory.

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.

This course is an introduction to stochastic calculus at the undergraduate level with applications to financial models. The emphasis is on computational and practical techniques. Topics include: Brownian motion; Ito's formula; stochastic differential equations; partial differential equations; Girsanov's theorem; simulation and finite difference numerical methods; implementation in Matlab; applications in finance.

This course introduces the unique features and mathematical models used in commodity (energy, metals, agricultural, etc.), and fixed income markets (bonds, rates, swaps, etc). While interest rate markets are a well-established field, commodities are an increasingly important asset class for financial institutions, policy makers and many investors. Mathematical finance approaches developed for stocks can be adapted to understand price behavior in these more complex markets. Topics include: energy prices; storable vs non-storable commodities; power and emissions; short rate models; forward curves and calibration; option pricing techniques.

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. Nonparametricegression and classification. Time series analysis: stationarity and classical linear models (AR, MA, ARMA, ..). Nonlinear and nonstationary 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.

This seminar is a continuation of ORF 509. Each student writes a report and presents research results. For doctoral students, the course must be completed one semester prior to taking the general examinations.

Course covers the pricing and hedging of advanced derivatives, including topics such as exotic options, greeks, interest rate derivatives and credit derivatives, as well as covering the basics of stochastic calculus necessary for finance. Designed for Masters students.

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 learning problems related to regression and classification. Topics covered include: Principle Component Analysis, nonparametric estimation, sparse regression, and Classification and Statistical learning

Graduate introduction to probability theory beginning with a review of measure and integration. Topics include random variables, expectation, characteristic functions, law of large numbers, central limit theorem, conditioning, martingales, Markov chains, and Poisson processes

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 will be introduced to C++, the weekly homework will involve writing C++ code, and the final project will also involve 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 in the-mean variance and expected utility sense, interest rate risk, credit risk, pricing and hedging of derivatives, risk measures, systemic risk.

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.

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.

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.

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: suprema of random processes; Gaussian and Rademacher inequalities; generic chaining; entropy and combinatorial dimensions; concentration of measure; functional, transportation cost, martingale inequalities; isoperimetry; Markov semigroups, mixing times, random fields; hypercontractivity; thresholds and influences; Stein's method; selected applications.

The course explores the modelling, measuring and managing of financial risk - i.e., differentiating between a good trade and a bad one. Emphasis will be placed on illustrating the modelling needs and strategies on the buy side vs. the sell side. Practical applications will be drawn from all asset classes. Convexity and time scale considerations will be addressed and risk management will be discussed with respect to RV and macro strategies. Coursework will focus on assessing risk/reward and identifying market opportunities. Entropy techniques and discretization issues will also be introduced. Guest speakers: experts from hedge funds and banks.