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dc.contributor.advisorDimitris Bertsimas.en_US
dc.contributor.authorPopescu, Ioanaen_US
dc.contributor.otherMassachusetts Institute of Technology. Operations Research Center.en_US
dc.date.accessioned2005-08-22T19:03:03Z
dc.date.available2005-08-22T19:03:03Z
dc.date.copyright1999en_US
dc.date.issued1999en_US
dc.identifier.urihttp://hdl.handle.net/1721.1/9526
dc.descriptionThesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics; and, (Ph.D.)--Massachusetts Institute of Technology, Operations Research Center, 1999.en_US
dc.descriptionIncludes bibliographical references (p. 144-151).en_US
dc.description.abstractThe unifying contribution of this thesis is to show that optimization is a very powerful tool that provides unexpected insights and impact on a variety of domains, such as probability, finance and revenue management. The thesis has two parts: In the first part, we use optimization models and techniques to derive optimal bounds for moment type problems in probability and finance. In the probability framework , we derive optimal inequalities for P(X E S), for a multivariate random variable X that has a given collection of moments, and an arbitrary set S. We provide a complete characterization of the problem of finding optimal bounds, from a complexity standpoint. We propose an efficient algorithm to compute tight bounds when S is a union of a polynomial number of convex sets, and up to second order moments of X are known. We show that it is NP-hard to obtain such bounds if the domain of X is Rn+, or if moments of third or higher order are given. Using convex optimization methods, we prove explicit tight bounds that generalize the classical Markov and Chebyshev inequalities, when the set S is convex. We examine implications to the law of large numbers, and the central limit theorem. In the finance framework, we investigate the applicability of such moment methods to obtain optimal bounds on financial quantities, when information about related instruments is available. We investigate the relation of option and stock prices just based on the no-arbitrage assumption, without assuming any model for the underlying price dynamics. We introduce convex optimization methods, duality and complexity theory to shed new light to this relation. We propose efficient algorithms for finding best possible bounds on option prices on multiple assets, based on the mean and variance of the underlying asset prices and their correlations and identify cases under which the derivation of such bounds is NP-hard. Conversely, given observable option prices, we provide best possible bounds on the moments of the underlying assets as well as prices of other options on the same asset. Our methods naturally extend for the case of transactions costs. The second part of this thesis applies dynamic and linear optimization methods to network revenue management applications. We investigate dynamic policies for allocating inventory to correlated, stochastic demand for multiple classes, in a network environment so as to maximize total expected revenues. We design a new efficient algorithm, based on approximate dynamic programming that provides structural insights into the optimal policy by using adaptive, non-additive bid-prices from a linear programming relaxation. Under mild restrictions on the demand process, our algorithm is asymptotically optimal as the number of periods in the time horizon increases, capacities being held fixed. In contrast, we prove that this is not true for additive bid-price mechanisms. We provide computational results that give insight into the performance of these algorithms, for several networks and demand scenarios. We extend these algorithms to handle cancellations and no-shows by incorporating overbooking decisions in the underlying mathematical programming formulation.en_US
dc.description.statementofresponsibilityby Ioana Popescu.en_US
dc.format.extent151 p.en_US
dc.format.extent9471447 bytes
dc.format.extent9471205 bytes
dc.format.mimetypeapplication/pdf
dc.format.mimetypeapplication/pdf
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsM.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582
dc.subjectMathematics.en_US
dc.subjectOperations Research Center.en_US
dc.titleApplications of optimization in probability, finance and revenue managementen_US
dc.typeThesisen_US
dc.description.degreePh.D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.contributor.departmentMassachusetts Institute of Technology. Operations Research Centeren_US
dc.identifier.oclc43855381en_US


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