Jarrow-Yildirim model for inflation: theory and applications

This thesis deals with inflation theory, focussing on the model of Jarrow & Yildirim, which is nowadays used when pricing inflation derivatives. After recalling main results about short and forward interest rate models, the dynamics of the main components of the market are derived. Then the most important inflation-indexed derivatives are explained (zero coupon swap, year-on-year, cap and floor), and their pricing proceeding is shown step by step. Calibration is explained and performed with a common method and an heuristic and non standard one. The model is enriched with credit risk, too, which allows to take into account the possibility of bankrupt of the counterparty of a contract. In this context, the general method of pricing is derived, with the introduction of defaultable zero-coupon bonds, and the Monte Carlo method is treated in detailed and used to price a concrete example of contract. Appendixes: A: martingale measures, Girsanov's theorem and the change of numeraire. B: some aspects of the theory of Stochastic Differential Equations; in particular, the solution for linear EDSs, and the Feynman-Kac Theorem, which shows the connection between EDSs and Partial Differential Equations. C: some useful results about normal distribution.

Transportation theory and planning

The aim of this research is to analyze the transport system and its subcomponents in order to highlight which are the design tools for physical and/or organizational projects related to transport supply systems. A characteristic of the transport systems is that the change of their structures can recoil on several entities, groups of entities, which constitute the community. The construction of a new infrastructure can modify both the transport service characteristic for all the user of the entire network; for example, the construction of a transportation infrastructure can change not only the transport service characteristics for the users of the entire network in which it is part of, but also it produces economical, social, and environmental effects. Therefore, the interventions or the improvements choices must be performed using a rational decision making approach. This approach requires that these choices are taken through the quantitative evaluation of the different effects caused by the different intervention plans. This approach becomes even more necessary when the decisions are taken in behalf of the community. Then, in order to understand how to develop a planning process in Transportation I will firstly analyze the transport system and the mathematical models used to describe it: these models provide us significant indicators which can be used to evaluate the effects of possible interventions. In conclusion, I will move on the topics related to the transport planning, analyzing the planning process, and the variables that have to be considered to perform a feasibility analysis or to compare different alternatives. In conclusion I will perform a preliminary analysis of a new transit system which is planned to be developed in New York City.

Knot theory and its applications

Nella tesi verranno presi in considerazione tre aspetti: si descriverà come la teoria dei nodi si sia sviluppata nel corso degli anni in relazione alle diverse scoperte scientifiche avvenute. Si potrà quindi subito avere una idea di come questa teoria sia estremamente connessa a diverse altre. Nel secondo capitolo ci si occuperà degli aspetti più formali di questa teoria. Si introdurrà il concetto di nodi equivalenti e di invariante dei nodi. Si definiranno diversi invarianti, dai più elementari, le mosse di Reidemeister, il numero di incroci e la tricolorabilità, fino ai polinomi invarianti, tra cui il polinomio di Alexander, il polinomio di Jones e quello di Kaufman. Infine si spiegheranno alcune applicazioni della teoria dei nodi in chimica, fisica e biologia. Sulla chimica, si definirà la chiralità molecolare e si mostrerà come la chiralità dei nodi possa essere utile nel determinare quella molecolare. In campo fisico, si mostrerà la relazione che esiste tra l'equazione di Yang-Baxter e i nodi. E in conclusione si mostrerà come modellare un importante processo biologico, la recombinazione del DNA, grazie alla teoria dei nodi.

Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Planck stars: theory and phenomenology

General Relativity (GR) is one of the greatest scientific achievements of the 20th century along with quantum theory. Despite the elegance and the accordance with experimental tests, these two theories appear to be utterly incompatible at fundamental level. Black holes provide a perfect stage to point out these difficulties. Indeed, classical GR fails to describe Nature at small radii, because nothing prevents quantum mechanics from affecting the high curvature zone, and because classical GR becomes ill-defined at r = 0 anyway. Rovelli and Haggard have recently proposed a scenario where a negative quantum pressure at the Planck scales stops and reverts the gravitational collapse, leading to an effective “bounce” and explosion, thus resolving the central singularity. This scenario, called Black Hole Fireworks, has been proposed in a semiclassical framework. The purpose of this thesis is twofold: - Compute the bouncing time by means of a pure quantum computation based on Loop Quantum Gravity; - Extend the known theory to a more realistic scenario, in which the rotation is taken into account by means of the Newman-Janis Algorithm.