Mathematics in Progress

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Displaying theses 1-10 of 395 total
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D. Wei
Master programme: Stochastics and Financial Mathematics October 30th, 2017
Institute: KdVI Research group: Stochastics and Financial Mathematics Graduation thesis Supervisor: A.V. (Arnoud) den Boer
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Decision based model selection for the newsvendor problem
In this thesis, the performance of a decision based model selection method is studied when it is applied to the newsvendor problem. The DBMS is a model selection method that selects models based on the performance of the decisions derived by the models rather than the goodness-of-fit of the underlying demand distribution. The expressions of the expected regret caused by DBMS are derived in the scenario cases where two models are available. The expressions are obtained by firstly computing the distributions of the decision and then calculating the regrets of the two decisions. The robustness measured by the maximum regret of a given decision, is discussed knowing partial information of the underlying model, and the maximum regret is shown to be achieved by a two-point distribution. The performance of DBMS is illustrated numerically when the underlying model assumes a two-point distribution. It is shown that DBMS performs better than the purely data-driven approach, based on the empirical distribution of the demands, when the performance is measured in terms of the expected cost.
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Scientific abstract (pdf 1K)   For more info or full text, mail to: A.V.denBoer@uva.nl

N. Disveld
Master programme: Mathematics October 25th, 2017
Institute: UvA / Other Research group: Korteweg-de Vries Institute for Mathematics Graduation thesis Supervisor: Jasper Stokman
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Nonsymmetric Interpolation Okounkov Polynomials
There exist (non-)symmetric interpolation polynomials that are connected to the famous (non-)symmetric Macdonald polynomials. With Laurent polynomials, the role of the (non-)symmetric Macdonald polynomials is being played by the (non-)symmetric Koornwinder polynomials. There exist symmetric interpolation Laurent polynomials that are connected to the symmetric Koornwinder polynomials, we give a new proof of this existence. Also, we give a definition of the non-symmetric interpolation Laurent polynomials that are connected to the non-symmetric Koornwinder polynomials and prove their existence.
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Scientific abstract (pdf 1K)   For more info or full text, mail to: j.v.stokman@uva.nl

P. Bongers
Master programme: Mathematics October 6th, 2017
Institute: KdVI Research group: Stochastics Graduation thesis Supervisor: prof. dr. R. Nunez Queija
Queuing on a continuous circle, a mathematical analysis of a void-avoiding optical fiber-loop
In this thesis, we consider a node where data packets arrive and have to be processed. A node can only process one job at the time, therefore excess jobs have to be placed in a buffer. For this purpose, an optical fiber-loop can be used. This fiber-loop creates a small time delay. When the job emerges from the fiber-loop, it can be transmitted or transferred into the fiber loop again. We begin our research by showing the relations between the fiber-model and queueing models. We describe the distribution of the queue content, the number of jobs that are in the system in steady state. In earlier literature, the mean and variance of the queue content were obtained. The goal of this master project is to understand this distribution better. We proceed by proving some properties of the model. Because direct analysis of the model is difficult, we present an approximation that makes the analysis of the model considerably easier. We show that the distribution of the number of jobs in the fiber-loop for the approximation is very similar to that of the original model, but is much easier to obtain.
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Scientific abstract (pdf 1K)   Full text (pdf 752K)

D. Mitkidou
Master programme: Stochastics and Financial Mathematics September 26th, 2017
Institute: KdVI Research group: Stochastics and Financial Mathematics Graduation thesis Supervisor: Asma Khedher
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Quadratic hedging strategies in affine models
This master thesis studies the problem of hedging European style options in incomplete markets. The most celebrated model which is often used for option pricing and hedging is the Black-Scholes model. This model imposes certain assumptions which allow to form perfect hedging strategies and therefore the risk involved in trading options is eliminated. The most signi ficant assumption is that volatility is constant over time. However, if we would like to take a more realistic view of the financial world, the latter assumption needs to be relaxed. In the context of this thesis, we consider stochastic volatility models, where the volatility is modelled as a stochastic process. In this class of models perfect hedging strategies do not exist. Our goal is to form such strategies that reduce the risk as much as possible. To approach this problem we use a method called quadratic hedging. Affine stochastic volatility models constitute one subclass of stochastic volatility models. They are of special interest due to their computational tractability which often allows to obtain closed-form solutions for pricing several options. This thesis uses the rich structural properties of affine models to obtain semiexplicit formulas for quadratic hedging strategies.
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Scientific abstract (pdf 1K)   Full text (pdf 568K)

P.A. van Reeuwijk
Master programme: Mathematics September 15th, 2017
Institute: KdVI Research group: Algebraic Geometry Graduation thesis Supervisor: Arno Kret
The Langlands-Kottwitz method for the modular curve
A moduli space is a space parametrising geometric objects; these are objects of primary interest in algebraic geometry. These spaces, however, are often very complicated. From a moduli space a zeta function can be construced, a complex function that, being an anayltic object, is supposed to be much easier to handle while still containing valuable information on the geometry and arithmetic of the moduli space. For this to work, the zeta function needs to satisfy a number of anaylic properties. We use the approach of Langlands and Kottwitz to work towards understanding the zeta fuction of moduli spaces of elliptic curves: we can use the special properties of these elliptic curves to explicitly count the number of points on their moduli spaces, leading to a better understanding of the properties of the zeta function.
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Scientific abstract (pdf 1K)   Full text (pdf 220K)

I.C. Bodin
Master programme: Stochastics and Financial Mathematics August 31st, 2017
Institute: UvA / Other Research group: Korteweg-de Vries Institute for Mathematics Graduation thesis Supervisor: Peter Spreij
Pricing and Hedging of Mortgage Option
This thesis provides different pricing methods for four mortgage options, namely the pipelineoption, the meeneemoption,the LtV-option and rentemiddeling. A theoretical approach is present for each option and also the results of numerical experiments are presented. Common techniques are Monte Carlo, trinomial trees and Lévy theory. Real data is used to calibrate the models to guarantee the most relevant results. These turn out to be in line with the result of the currently used behaviour models in case of European options. American options are differently priced and the bank should be aware of the possible consequences.
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Scientific abstract (pdf 1K)   Full text (pdf 6498K)

M.J.M. Derksen
Master programme: Stochastics and Financial Mathematics August 29th, 2017
Institute: KdVI Research group: Stochastics and Financial Mathematics Graduation thesis Supervisor: Peter Spreij
Pricing of Contingent Convertible Bonds
In this thesis different pricing models are studied for the pricing of Contingent Convertible bonds (CoCos). These are special type of bonds, which convert into equity or are written down, when the capital of the issuing bank becomes too low. In this way, outstanding debt is reduced and capital is raised, to strengthen the capital position of the bank. In practice, this conversion is typically triggered by the capital ratio of the issuing bank falling below some threshold or by a regulator calling for conversion. However, in all of the existing pricing models the conversion of CoCos is triggered by a market value, like a stock price, falling below some threshold. In this thesis a model is proposed, in which the market cannot observe the true asset value process, but it only has access to noisy accounting reports, which are only published at discrete moments in time. In this way, the price of CoCos can only be based on the information from the accounting reports, not on the true asset process.
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Scientific abstract (pdf 1K)   Full text (pdf 811K)

I.S. Liesker
Master programme: Stochastics and Financial Mathematics August 22nd, 2017
Institute: KdVI Research group: Stochastics and Financial Mathematics Graduation thesis Supervisor: Peter Spreij
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Affine and quadratic interest rate models: A theoretical and empirical comparison
In the financial world people try to speculate about the financial market. There are many variables that are unknown and that one wants to describe by, for example, stochastic models. These models help to get insight in the financial variables and are sometimes even used to predict the future development of the variable in order to do proper investments or protect themselves against risk. The latter, in form of interest rate risk modeling, is studied in this thesis. One of the popular interest rates models is the affine model. Affine models are becoming increasingly popular due to their analytical and computational tractability. Affine processes have a nice pricing formula for multiple financial products. Quadratic processes are, to some extent, an extension of affine models and have similar properties as affine models. This thesis compares these affine and quadratic models on a theoretical and an empirical level. For the theoretical level, the mathematics of affine and quadratic interest rate models is explained. For both affine and quadratic models analytical ('nice') formulas for some financial products are provided using admissible parameters and Riccati equations. Also, using the analytical bond prices, a small empirical comparison is performed where some computational examples are discussed.
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Scientific abstract (pdf 1K)   Full text (pdf 3861K)

R.Q. Riksen
Master programme: Stochastics and Financial Mathematics August 22nd, 2017
Institute: KdVI Research group: Stochastics and Financial Mathematics Graduation thesis Supervisor: Peter Spreij
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Using Artificial Neural Networks in the Calculation of Mortgage Prepayment Risk
A client with a mortgage loan has the possibility to pay back part of his mortgage before the end of the contract. Because this poses a risk to the bank due to the loss of future interest payments, it is very important to predict the probability that a client will prepay on his mortgage. There are many parameters that can influence these mortgage prepayments in a complicated way. Artificial neural networks are used as approximators. A network consists of many connected nodes, that are grouped into layers. Each node takes a weighted sum of all the input it receives, applies a certain function to it and sends it on to all neurons in the next layer. The key to making a neural network approximate the target function, is to make it `learn' the correct weights. It gets to see a lot of input values and makes predictions. If the prediction was incorrect, all weights are changed a little in the direction that will make the network give a better prediction next time. This way, the network learns by making mistakes. In this thesis at ABN AMRO, we explore how we can use artificial neural networks to predict prepayment behaviour.
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Scientific abstract (pdf 1K)   Full text (pdf 1396K)

C. Groot
Master programme: Mathematics August 16th, 2017
Institute: KdVI Research group: Stochastics Graduation thesis Supervisor: S.G. Cox
Wellposedness of Stochastic Differential Equations in Infinite Dimensions
We investigate the wellposedness of stochastic differential equations in infinite dimensions, following the variational approach given in Liu and Röckner (Stochastic Partial Differential Equations: An Introduction, Springer, 2015). We look at the existence and uniqueness of (variational) solutions to stochastic differential equations driven by an infinite-dimensional standard cylindrical Wiener process. The results we prove require some preliminary knowledge on Bochner integrals and probability and martingale theory in infinite-dimensional Banach spaces amongst other results. We will also introduce the stochastic integral with respect to a Q-Wiener process and a standard cylindrical Wiener process. After that, we sketch the setting for the main result: we discuss the Gelfand triple and sketch the general setting of the existence and uniqueness result. We impose conditions on the coefficients of the stochastic differential equation, namely hemicontinuity, boundedness, coercivity and weak monotonicity. The proof of existence relies on the existence of strong solutions for finite-dimensional SDEs and weak convergence results. The uniqueness follows from an integration-by-parts argument.
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Scientific abstract (pdf 50K)   Full text (pdf 1167K)

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