Mathematics Colloquia and Seminars

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It is often the case in various branches of applied mathematics that one wishes to solve optimization problems involving parameters that are estimated from data. In that setting, it is natural to ask the following question: what is the relationship between the solution of the optimization problem with estimated parameters (i.e the sample version) and the solution we would get were we to know the actual value of the parameters (the population version)? An example of particular interest is the classical Markowitz portfolio optimization problem in finance. I will discuss some of these questions in the ``large n, large p" setting that is now quite often considered in high-dimensional statistics and is typical in random matrix theory. In particular, I will discuss results highlighting the fact that the dimensionality of the data implies risk underestimation in the ``stylized-finance" problem considered here. I will also discuss robustness and lack thereof of the conclusions to various distributional assumptions. No prior knowledge of finance or random matrix theory will be assumed.