# Operations Research and Financial Engineering

## Overview

*Foundations*

The ORFE program places a strong emphasis on mathematical and computational tools. Students in ORFE develop a unique set of skills that build upon a solid foundation in probability, statistics, and optimization.

*Applications*

The theoretical foundations of ORFE are of central importance in many complex problems in engineering and science. Students and faculty in ORFE work in a broad range of application areas, such as finance, energy, health, risk analysis, biostatistics, genomics, machine learning, operations research, stochastic networks, signal and image processing, automated vehicle control systems, optimal design of engineered systems, robotics, astrophysics, and homeland security.

*Impact*

Graduates of the Ph.D. program work in academia, research organizations, and industry. Many ORFE graduates hold faculty positions at top universities.

The department offers two degree programs: the Doctor of Philosophy (Ph.D.) in Operations Research and Financial Engineering, and a Master of Science in Engineering (M.S.E.). These programs provide a great deal of flexibility for students in designing individual plans of study and research according to their needs and interests. The department is a major participant in the Master of Finance (M.Fin.) program offered through the Bendheim Center for Finance.

## Applying

- All applicants are required to submit a GRE general test. A mathematics subject test is strongly recommended.
- M.S.E. applicants are required to have the endorsement of a faculty member who is willing to supervise them prior to submitting an application.
- M.S.E. applicants are required to submit a Statement of Financial Resources.

## Ph.D.

The Ph.D. program is formulated to prepare students for research and teaching. The aim of the program is to provide a strong disciplinary background in at least one of the core areas of research in the department. The emphasis is on the theoretical foundations, mathematical models, and computational issues in practical problem-solving. Current teaching and research activities include stochastic analysis, mathematical statistics, machine learning, analysis of big data, linear and nonlinear optimization, stochastic optimization, convex analysis, stochastic networks, queueing theory, mathematical and computational finance, PDE methods, stochastic control, dynamic game theory, and financial econometrics. Application areas of current interest to faculty include finance, energy, health, biostatistics, genomics, robotics, social networks, and astrophysics.

In the first year of graduate study, students must take all six core courses. By the end of the first year, students are expected to narrow the area of doctoral research and choose an appropriate adviser. Second-year students are required to complete a qualifying examination, undertake advanced coursework, research projects, and prepare for the general examination. The general examination is normally taken at the end of the second year.

Beyond the general examination, the completion of a dissertation usually takes two to three years. Upon acceptance of the dissertation by the department and the Graduate School, candidates for the Ph.D. stand for the final public oral examination, which is primarily a defense of the dissertation.

In consultation with the director of graduate studies, students develop a specific course plan. During the first year, students complete six courses that emphasize the foundations of the program, probability, statistics, and optimization. Students take a total of at least ten courses while enrolled: the six core courses during the first year, two directed research courses andtwo additional advanced-level courses in the second year.

The following six core courses are required:

- ORF 522 Linear and Nonlinear Optimization
- ORF 523 Convex and Conic Optimization
- ORF 524 Statistical Theory and Methods
- ORF 525 Statistical Foundations of Data Science
- ORF 526 Probability Theory
- ORF 527 Stochastic Calculus

In addition, at least two advanced courses and two semesters of directed research (ORF 509 and ORF 510) are completed under the direction of a faculty adviser in the student's area of interest by the end of the second year in preparation for the general examination.

*Research Adviser*

Students ordinarily match with an adviser by the end of the first year or study in order to begin research in preparation for the general examination. No student may enter the second year without a research adviser of record. Students are expected to identify a faculty adviser in the ORFE department. If a student receives permission to work with a faculty advisor who is not a member of the ORFE department, the student must also have a co-adviser who is on the ORFE faculty and is seriously involved in co-advising the student.

*Qualifying Examination*

Each student must satisfy qualifying requirements. Qualifying exams are offered in September of the student’s second year.

A student who obtained a grade of A- or better in four of the six required core classes will be exempt from the September qualifying exams. If this is not the case, the student will meet with the DGS, who will determine which exams need to be taken in September to satisfy the requirements. The optimization exams are based on ORF 522 and ORF 523. The probability exams are based on ORF 526 and ORF 527. The statistics exams are based on ORF 524 and ORF 525.

The results of the qualifying exam are determined by a vote of the faculty. Students who fail must transfer out of the Ph.D. program. There is no option to retake the exam.

ORFE students take the general exam in April or May of their second year. In order to be eligible to stand for the general examination, students must have completed the required core courses, met the qualifying examination requirements, have taken and passed ORF 509, have taken or are currently enrolled in ORF 510, and have received a B+ or higher in two additional advanced courses at the 500 level. The student must also show adequate progress on research and an acceptable level of understanding of his or her area of specialization.

The general examination consists of two parts, a written and oral component, both covering the student's area of specialty. The written component is completed by submitting a written report on the research conducted in ORF 509-510. It is due one week before the exam takes place. The report serves as the basis for the student’s presentation. The oral component is completed by giving a presentation on the research presented in the comprehensive written report. The oral exam may be up to 3 hours in length.

As part of the oral examination, the general examination committee may ask questions related to the research as well as the student’s area of specialization.

For each student, an examining committee is selected by the student and adviser. In addition to the adviser, the committee consists of two ORFE faculty or affiliated faculty and must be approved by the Director of Graduate Studies. The committee will administer the oral exam, evaluate the student’s performance in research and overall knowledge of the student's field, and make a recommendation to the department faculty. A departmental faculty vote determines the final outcome.

The Master of Arts (M.A.) degree is normally an incidental degree on the way to full Ph.D. candidacy and is earned after a student successfully passes the general examination. It may also be awarded to students who, for various reasons, leave the Ph.D. program, provided that these requirements have been met.

Please note, students admitted to the Ph.D. program who do not wish to complete the program may be considered for an M.S.E. degree with approval from the department and the Graduate School. Ph.D. students who have already been awarded the incidental M.A. are not eligible to earn an M.S.E.

Upon completion and acceptance of the dissertation by the department, the candidate will be admitted to the final public oral (FPO) examination.

The committee of examiners for the FPO must consist of no fewer than two current ORFE faculty members, and the 2nd thesis reader should also be a current ORFE faculty member.

The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.

## M.S.E.

The ORFE department is primarily geared towards educating students pursuing a Ph.D. in the field. As such, the Master of Science in Engineering (M.S.E.) program in ORFE also has a strong research focus, as reflected in the requirement of a thesis and full-time study for two academic years. Students enrolled in this program are eligible for financial support in the form of research or teaching assistantships if such funds are available.

The admission rate for the M.S.E. degree is very low. Admission is based on the qualifications of the applicant and requires support of at least one faculty member who expresses an interest in supervising the applicant. Applicants interested in an M.S.E. degree from ORFE are urged to identify and contact a faculty member in whose area of research they would like to work.

Applicants who are primarily interested in a master's degree in finance should apply for the Master in Finance Program at the Bendheim Center for Finance. The School of Engineering website provides more information regarding the Master of Science in Engineering program.

The course requirements are fulfilled by successfully completing ten one-semester courses approved by the department, two of which are required research courses (ORF 509 and 510).

The M.S.E. program has a strong research focus reflected in the requirement of a thesis. Upon completion and acceptance of the thesis by the department, the candidate will be admitted to the final defense, administered by at least two faculty members.

## M.Eng.

The Master of Engineering (M.Eng.) is a coursework-based master's degree offered, in 2021-22, to current ORFE Seniors. Candidates for the M.Eng. degree will be enrolled full time throughout the 10-month academic year. No research or thesis is required. Any class (or its graduate equivalent) counted towards the Princeton undergraduate degree cannot be counted for the M.Eng. degree.

Candidates for the M.Eng. degree must successfully complete at least eight approved courses. A minimum of six of these eight courses must be technical, having their primary listing in a department or a program within the natural sciences or engineering. A minimum of four of these six courses must be chosen from graduate offerings in the Department of Operations Research & Financial Engineering; options include any of the following six core courses for the Ph.D. degree. Students must have a “B” (3.0) average or better at the time they complete the program requirements in order to receive the degree.

ORF 522- Linear & Nonlinear Optimization

ORF 523- Convex and Conic Optimization

ORF 524- Statistical Theory and Methods

ORF 525- Statistical Foundations of Data Science

ORF 526- Probability Theory

ORF 527- Stochastic Calculus

As well as several graduate-level ORFE electives

ORF 504- Financial Econometrics

ORF 505- Statistical Analysis of Financial Data

ORF 515- Asset Pricing II: Stochastic Calculus and Advanced Derivatives

ORF 531- Computational Finance in C++

ORF 535- Financial Risk Management

ORF 538- PDE Methods in Financial Mathematics

ORF 542- Stochastic Control and Stochastic Differential Games

ORF 545- High Frequency Markets

ORF 569- Special Topics in Statistics, Operations Research and Financial Engineering

ORF 570- Special Topics in Statistics, Operations Research and Financial Engineering

Other approved classes include:

ORF 401- Electronic Commerce

ORF 407- Fundamentals of Queueing Theory

ORF 409- Introduction to Monte Carlo Simulation

ORF 418- Sequential Decision Analytics and Modeling

ORF 435- Financial Risk and Wealth Management

ORF 455- Energy & Commodities Markets

ORF 473- Special Topics in Operations Research and Financial Engineering

ORF 474- Special Topics in Operations Research and Financial Engineering

Courses of potential interest in other departments:

APC 503- Analytical Techniques in Differential Equations

COS 402- Machine Learning and Artificial Intelligence

COS 423- Theory of Algorithms

COS 485- Neural Networks: Theory and Application

COS 511- Theoretical Machine Learning

COS 521- Advanced Algorithm Design

COS 528- Data Structures and Graph Algorithms

COS 534- Fairness in Machine Learning

ECO 418- Strategy and Information

ECO 462- Portfolio Theory and Asset Management

ECO 464- Corporate Restructuring

ECO 466- Fixed Income: Models and Applications

ECO 467- Institutional Finance

ECO 517- Econometric Theory I

ECO 518- Econometric Theory II

ECO 525- Financial Economics I

ECO 526- Financial Economics II

ELE 525- Random Processes in Information Systems

ELE 535- Machine Learning and Pattern Recognition

ELE 538B- Information Sciences and Systems Large Scale Optimization for Data Science

MAE 546- Optimal Control and Estimation

MAT 522/APC 522- Introduction to PDE

MAT 527- Topics in Differential Equations: Global Solutions of Nonlinear Evolutions

MAT 572/APC 572- Introduction to Combinatorial Optimization

MAT 587- Topics in Ergodic Theory: Introduction to Ergodic Theory

MAT 589- Topics in Probability, Statistics and Dynamics: Modern Discrete Probability Theory

PHY 521/MAT 597- Introduction to Mathematical Physics

SPI 509- Generalized Linear Statistical Models

## Faculty

### Chair

- Ronnie Sircar

### Director of Graduate Studies

- Mykhaylo Shkolnikov

### Director of Undergraduate Studies

- Alain L. Kornhauser

### Professor

- Amir Ali Ahmadi
- René A. Carmona
- Matias D. Cattaneo
- Jianqing Fan
- Alain L. Kornhauser
- William A. Massey
- John M. Mulvey
- Warren B. Powell
- Ronnie Sircar
- Mete Soner
- Robert J. Vanderbei

### Associate Professor

- Mykhaylo Shkolnikov
- Ramon van Handel

### Assistant Professor

- Boris Hanin
- Jason Matthew Klusowski
- Miklos Z. Racz
- Bartolomeo Stellato
- Ludovic Tangpi

### Associated Faculty

- Yacine Aït-Sahalia, Economics
- Markus K. Brunnermeier, Economics
- Maria Chudnovsky, Mathematics
- Weinan E, Mathematics
- Sanjeev R. Kulkarni, Dean of the Faculty
- H. Vincent Poor, Electrical Engineering
- Paul Seymour, Mathematics
- Yakov G. Sinai, Mathematics
- John D. Storey, Integrative Genomics
- Wei Xiong, Economics

### Lecturer

- Margaret J. Holen
- Mathieu Lauriere
- Daniel C. Scheinerman
- Shen Shen

### Visiting Professor

- Johannes K. Ruf

### Visiting Assistant Professor

- Ricardo Pereira Masini

### Visiting Lecturer

- Robert F. Almgren
- Michael C. Coulon
- Michael G. Sotiropoulos

## Courses

Permanent Courses

Courses listed below are graduate-level courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and one-time-only graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduate-level independent reading and research courses, which may be approved by the Graduate School for individual students.