Essays on strategic trading

This dissertation discusses various aspects of strategic trading using both analytical modeling and numerical methods. Strategic trading, in short, encompasses models of trading, most notably models of optimal execution and portfolio selection, in which one seeks to rigorously consider various---both explicit and implicit---costs stemming from the act of trading itself. The strategic trading approach, rooted in the market microstructure literature, contrasts with many classical finance models in which markets are assumed to be frictionless and traders can, for the most part, take prices as given. Introducing trading costs to dynamic models of financial markets tend to complicate matters. First, the objectives of the traders become more nuanced since now overtrading leads to poor outcomes due to increased trading costs. Second, when trades affect prices and there are multiple traders in the market, the traders start to behave in a more calculated fashion, taking into account both their own objectives and the perceived actions of others. Acknowledging this strategic behavior is especially important when the traders are asymmetrically informed. These new features allow the models discussed to better reflect aspects real-world trading, for instance, intraday trading patterns, and enable one to ask and answer new questions, for instance, related to the interactions between different traders. To efficiently analyze the models put forth, numerical methods must be utilized. This is, as is to be expected, the price one must pay from added complexity. However, it also opens an opportunity to have a closer look at the numerical approaches themselves. This opportunity is capitalized on and various new and novel computational procedures influenced by the growing field of numerical real algebraic geometry are introduced and employed. These procedures are utilizable beyond the scope of this dissertation and enable one to sharpen the analysis of dynamic equilibrium models.Tämä väitöskirja käsittelee strategista kaupankäyntiä hyödyntäen sekä analyyttisiä että numeerisia menetelmiä. Strategisen kaupankäynnin mallit, erityisesti optimaalinen kauppojen toteutus ja portfolion valinta, pyrkivät tarkasti huomioimaan kaupankäynnistä itsestään aiheutuvat eksplisiittiset ja implisiittiset kustannukset. Tämä erottaa strategisen kaupankäynnin mallit klassisista kitkattomista malleista. Kustannusten huomioiminen rahoitusmarkkinoiden dynaamisessa tarkastelussa monimutkaistaa malleja. Ensinnäkin kaupankävijöiden tavoitteet muuttuvat hienovaraisemmiksi, koska liian aktiivinen kaupankäynti johtaa korkeisiin kaupankäyntikuluihin ja heikkoon tuottoon. Toiseksi oletus siitä, että kaupankävijöiden valitsemat toimet vaikuttavat hintoihin, johtaa pelikäyttäytymiseen silloin, kun markkinoilla on useampia kaupankävijöitä. Pelikäyttäytymisen huomioiminen on ensiarvoisen tärkeää, mikäli informaatio kaupankävijöiden kesken on asymmetristä. Näiden piirteiden johdosta tässä väitöskirjassa käsitellyt mallit mahdollistavat abstrahoitujen rahoitusmarkkinoiden aiempaa täsmällisemmän tarkastelun esimerkiksi päivänsisäisen kaupankäynnin osalta. Tämän lisäksi mallien avulla voidaan löytää vastauksia uusiin kysymyksiin, kuten esimerkiksi siihen, millaisia ovat kaupankävijöiden keskinäiset vuorovaikutussuhteet dynaamisilla markkinoilla. Monimutkaisten mallien analysointiin hyödynnetään numeerisia menetelmiä. Tämä avaa mahdollisuuden näiden menetelmien yksityiskohtaisempaan tarkasteluun, ja tätä mahdollisuutta hyödynnetään pohtimalla laskennallisia ratkaisuja tuoreesta numeerista reaalista algebrallista geometriaa hyödyntävästä näkökulmasta. Väitöskirjassa esitellyt uudet laskennalliset ratkaisut ovat laajalti hyödynnettävissä, ja niiden avulla on mahdollista terävöittää dynaamisten tasapainomallien analysointia

Polynomial systems : graphical structure, geometry, and applications

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 199-208).Solving systems of polynomial equations is a foundational problem in computational mathematics, that has several applications in the sciences and engineering. A closely related problem, also prevalent in applications, is that of optimizing polynomial functions subject to polynomial constraints. In this thesis we propose novel methods for both of these tasks. By taking advantage of the graphical and geometrical structure of the problem, our methods can achieve higher efficiency, and we can also prove better guarantees. Various problems in areas such as robotics, power systems, computer vision, cryptography, and chemical reaction networks, can be modeled by systems of polynomial equations, and in many cases the resulting systems have a simple sparsity structure. In the first part of this thesis we represent this sparsity structure with a graph, and study the algorithmic and complexity consequences of this graphical abstraction. Our main contribution is the introduction of a novel data structure, chordal networks, that always preserves the underlying graphical structure of the system. Remarkably, many interesting families of polynomial systems admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Our methods outperform existing techniques by orders of magnitude in applications from algebraic statistics and vector addition systems. We then turn our attention to the study of graphical structure in the computation of matrix permanents, a classical problem from computer science. We provide a novel algorithm that requires Õ(n 2[superscript w]) arithmetic operations, where [superscript w] is the treewidth of its bipartite adjacency graph. We also investigate the complexity of some related problems, including mixed discriminants, hyperdeterminants, and mixed volumes. Although seemingly unrelated to polynomial systems, our results have natural implications on the complexity of solving sparse systems. The second part of this thesis focuses on the problem of minimizing a polynomial function subject to polynomial equality constraints. This problem captures many important applications, including Max-Cut, tensor low rank approximation, the triangulation problem, and rotation synchronization. Although these problems are nonconvex, tractable semidefinite programming (SDP) relaxations have been proposed. We introduce a methodology to derive more efficient (smaller) relaxations, by leveraging the geometrical structure of the underlying variety. The main idea behind our method is to describe the variety with a generic set of samples, instead of relying on an algebraic description. Our methods are particularly appealing for varieties that are easy to sample from, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties we can take advantage of the tools from numerical algebraic geometry. Optimization problems from applications usually involve parameters (e.g., the data), and there is often a natural value of the parameters for which SDP relaxations solve the (polynomial) problem exactly. The final contribution of this thesis is to establish sufficient conditions (and quantitative bounds) under which SDP relaxations will continue to be exact as the parameter moves in a neighborhood of the original one. Our results can be used to show that several statistical estimation problems are solved exactly by SDP relaxations in the low noise regime. In particular, we prove this for the triangulation problem, rotation synchronization, rank one tensor approximation, and weighted orthogonal Procrustes.by Diego Cifuentes.Ph. D

A Geometric Approach to the Projective Tensor Norm

The main focus of this thesis is on the projective norm on finite-dimensional real or complex tensor products. There are various mathematical subjects with relations to the projective norm. For instance, it appears in the context of operator algebras or in quantum physics. The projective norm on multipartite tensor products is considered to be less accessible. So we use a method from convex algebraic geometry to approximate the projective unit ball by convex supersets, so-called theta bodies. For real multipartite tensor products we obtain theta bodies which are close to the projective unit ball, leading to a generalisation of the Schmidt decomposition. In a second step the method is applied to complex tensor products, in a third step to separable states. In a more general context, the projective norm can be related to binomial ideals, especially to so-called Hibi relations. In this respect, we also focus on a generalisation of the projective unit ball, here called Hibi body, and its theta bodies. It turns out that many statements also hold in this general context

An introduction to constructive algebraic analysis and its applications

This text is an extension of lectures notes I prepared for les Journées Nationales de Calcul Formel held at the CIRM, Luminy (France) on May 3-7, 2010. The main purpose of these lectures was to introduce the French community of symbolic computation to the constructive approach to algebraic analysis and particularly to algebraic D-modules, its applications to mathematical systems theory and its implementations in computer algebra systems such as Maple or GAP4. Since algebraic analysis is a mathematical theory which uses different techniques coming from module theory, homological algebra, sheaf theory, algebraic geometry, and microlocal analysis, it can be difficult to enter this fascinating new field of mathematics. Indeed, there are very few introducing texts. We are quickly led to Björk's books which, at first glance, may look difficult for the members of the symbolic computation community and for applied mathematicians. I believe that the main issue is less the technical difficulty of the existing references than the lack of friendly introduction to the topic, which could have offered a general idea of it, shown which kind of results and applications we can expect and how to handle the different computations on explicit examples. To a very small extent, these lectures notes were planned to fill this gap, at least for the basic ideas of algebraic analysis. Since, we can only teach well what we have clearly understood, I have chosen to focus on my work on the constructive aspects of algebraic analysis and its applications.Ce texte est une extension des notes de cours que j'ai préparés pour les les Journées Nationales de Calcul Formel qui ont eu lieu au CIRM, Luminy (France) du 3 au 7 Mai 2010. Le but principal de ce cours était d'introduire la communauté française du calcul formel à l'analyse algébrique constructive, et particulièrement à la théorie des D-modules algébriques, à ses applications à la théorie mathématique des systèmes et à ses implantations dans des logiciels de calcul formel tels que Maple ou GAP4. Parce que l'analyse algébrique est une théorie mathématique qui utilise différentes techniques venant de la théorie des modules, de l'algèbre homologique, de la théorie des faisceaux, de la géométrie algébrique et de l'analyse microlocale, il peut être difficile d'entrer dans ce domaine, nouveau et fascinant, des mathématiques. En effet, il existe peu de textes introductifs. Nous sommes rapidement conduits aux livres de Björk qui, à première vue, peuvent sembler difficiles aux membres de la communauté de calcul formel et aux mathématiciens appliqués. Je pense que le problème vient moins de la difficulté technique de la littérature existante que du manque d'introductions pédagogiques qui donnent une idée globale du domaine, montrent quels types de résultats et d'applications on peut attendre et qui développent les différents calculs à mener sur des exemples explicites. A leur humble niveau, ces notes de cours ont pour but de combler ce manque, tout du moins en ce qui concerne les idées de base de l'analyse algébrique. Puisque l'on ne peut enseigner bien que les choses que l'on a bien comprises, j'ai choisi de restreindre cette introduction à mes travaux sur les aspects constructifs de l'analyse algébrique et sur ses applications