Financial mathematics concerns mathematical models and problems arising in financial markets and applies tools from probability, optimization, stochastic analysis and statistics. Specific areas of research include risk management, pricing and hedging in incomplete markets, stochastic volatility models, markets with transaction costs, energy markets, credit risk, portfolio optimization, utility indifference valuation, and stochastic differential games.
Machine learning emerges from the need to design algorithms that are capable of learning from data how to make accurate predictions and decisions. Such problems arise in a variety of "big data" domains such as finance, genomics, information technologies and neuroscience. Research at ORFE ranges from the design of large-scale machine learning algorithms to their mathematical analysis.
Operations research combines the use of optimization, probability and statistics to solve problems in contextual domains such as business, energy systems, health services, financial services, telecommunications and transportation. Active areas of research often work at the intersection of these disciplines, such as the use of optimization in the estimation of large scale statistical models, optimal collection of information, and stochastic optimization.
Optimization is concerned with the analysis and algorithmic aspects of maximizing or minimizing an objective function subject to constraints, often in complex problems in high dimension. Active research areas in ORFE include interior-point methods, the parametric simplex method, stochastic optimization, and convex analysis. Applications of interest span from portfolio optimization to engineering applications such as optimal control, optical design, and machine learning.
Probability theory is the mathematical description of random phenomena. Probability plays an increasingly important role in almost all areas of engineering and science. Probability research in ORFE ranges from theoretical to applied, with particular emphasis on stochastic analysis and its applications in various areas including financial mathematics, stochastic networks and queueing, signal and image processing, and stochastic control.
Statistical research at ORFE is focused on the design of new statistical methods and their mathematical analysis. Specific areas of research include high-dimensional statistics, nonparametric statistics, nonlinear time series, sequential learning, combinatorial statistics, longitudinal and functional data analysis, and robust statistics. Areas of application span a variety of scientific domains including risk management, econometrics, machine learning, computational biology and biostatistics.