Thursday, August 31, 2017
Thursday, April 6, 2017
Thursday, March 30, 2017
Thursday, March 16, 2017
Thursday, March 2, 2017
Thursday, February 23, 2017
Wednesday, February 22, 2017
Thursday, February 9, 2017
Thursday, February 2, 2017
Thursday, January 26, 2017
Thu, 01/26/2017 - 3:05pm, Skiles006
Vladimir Koltchinskii, Georgia Tech Organizer: Christian Houdré
We study the problem of estimation of a linear functional of the eigenvector of covariance operator that corresponds to its largest eigenvalue (the first principal component) based on i.i.d. sample of centered Gaussian observations with this covariance. The problem is studied in a dimension-free framework with its complexity being characterized by so called "effective rank" of the true covariance. In this framework, we establish a minimax lower bound on the mean squared error of estimation of linear functional and construct an asymptotically normal estimator for which the bound is attained. The standard "naive" estimator (the linear functional of the empirical principal component) is suboptimal in this problem. The talk is based on a joint work with Richard Nickl.
Thursday, January 19, 2017
Thu, 01/19/2017 - 3:05pm, Skiles 006
Dave Goldberg, ISyE, GaTech Organizer: Christian Houdré
Demand forecasting plays an important role in many inventory control problems. To mitigate the potential harms of model misspecification, various forms of distributionally robust optimization have been applied. Although many of these methodologies suffer from the problem of time-inconsistency, the work of Klabjan et al. established a general time-consistent framework for such problems by connecting to the literature on robust Markov decision processes. Motivated by the fact that many forecasting models exhibit very special structure, as well as a desire to understand the impact of positing different dependency structures, in this talk we formulate and solve a time-consistent distributionally robust multi-stage newsvendor model which naturally unifies and robustifies several inventory models with demand forecasting. In particular, many simple models of demand forecasting have the feature that demand evolves as a martingale (i.e. expected demand tomorrow equals realized demand today). We consider a robust variant of such models, in which the sequence of future demands may be any martingale with given mean and support. Under such a model, past realizations of demand are naturally incorporated into the structure of the uncertainty set going forwards. We explicitly compute the minimax optimal policy (and worst-case distribution) in closed form, by combining ideas from convex analysis, probability, and dynamic programming. We prove that at optimality the worst-case demand distribution corresponds to the setting in which inventory may become obsolete at a random time, a scenario of practical interest. To gain further insight, we prove weak convergence (as the time horizon grows large) to a simple and intuitive process. We also compare to the analogous setting in which demand is independent across periods (analyzed previously by Shapiro), and identify interesting differences between these models, in the spirit of the price of correlations studied by Agrawal et al. This is joint work with Linwei Xin, and the paper is available on arxiv at https://arxiv.org/abs/1511.09437v1
Monday, January 9, 2017
Mon, 01/09/2017 - 3:05pm, Skiles 006
Franck Maunoury, Université Pierre et Marie Curie Organizer: Christian Houdré
We consider permanental and multivariate negative binomial distributions. We give sim- ple necessary and sufficient conditions on their kernel for infinite divisibility, without symmetry hypothesis. For existence of permanental distributions, conditions had been given by Kogan and Marcus in the case of a 3 × 3 matrix kernel: they had showed that such distributions exist only for two types of kernels (up to diagonal similarity): symmet- ric positive-definite matrices and inverse M-matrices. They asked whether there existed other classes of kernels in dimensions higher than 3. We give an affirmative answer to this question, by exhibiting (in any finite dimension higher than 3) a family of matrices which are kernels of permanental distributions but are neither symmetric, nor inverse M-matrices (up to diagonal similarity). Analog properties (by replacing inverse M-matrices by entrywise non-negative matrices) are given for multivariate negative binomial distribu- tions. These notions are also linked with the study of inverse power series of determinant. This is a joint work with N. Eisenbaum.
Thursday, November 3, 2016
Thu, 11/03/2016 - 3:05pm, Skiles 006
Exact oracle inequalities for regression have been extensively studied in statistics and learning theory over the past decade. In the case of a misspecified model, the focus has been on either parametric or convex classes. We present a new estimator that steps outside of the model in the non-convex case and reduces to least squares in the convex case. To analyze the estimator for general non-parametric classes, we prove a generalized Pythagorean theorem and study the supremum of a negative-mean stochastic process (which we term the offset Rademacher complexity) via the chaining technique.(joint work with T. Liang and K. Sridharan)
Thursday, October 27, 2016
Short-Time Expansions for Call Options on Leveraged ETFs Under Exponential Levy Models with Local Volatility
Thu, 10/27/2016 - 3:05pm, Skiles 006
R. Gong, Illinois Institute of Technology Organizer: Christian Houdré
In this talk, we consider the small-time asymptotics of options on a Leveraged Exchange-Traded Fund (LETF) when the underlying Exchange Traded Fund (ETF) exhibits both local volatility and jumps of either finite or infinite activity. Our main results are closed-form expressions for the leading order terms of off-the-money European call and put LETF option prices, near expiration, with explicit error bounds. We show that the price of an out-of-the-money European call on a LETF with positive (negative) leverage is asymptotically equivalent, in short-time, to the price of an out-of-the-money European call (put) on the underlying ETF, but with modied spot and strike prices. Similar relationships hold for other off-the-money European options. In particular, our results suggest a method to hedge off-the-money LETF options near expiration using options on the underlying ETF. Finally, a second order expansion for the corresponding implied volatilities is also derived and illustrated numerically. This is the joint work with J. E. Figueroa-Lopez and M. Lorig.
Thursday, September 29, 2016
Thu, 09/29/2016 - 3:05pm, Skiles 006
We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. This is joint work with Marek Biskup.