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.

Lateral tires characterization: testing, analytical models and applications for aeronautic purpose

The lateral characteristics of tires in terms of lateral forces as a function of sideslip angle is a focal point in the prediction of ground loads and ground handling aircraft behavior. However, tests to validate such coefficients are not mandatory to obtain Aircraft Type Certification and so they are not available for ATR tires. Anyway, some analytical values are implemented in ATR calculation codes (Flight Qualities in-house numerical code and Loads in-house numerical code). Hence, the goal of my work is to further investigate and validate lateral tires characteristics by means of: exploitation and re-parameterization of existing test on NLG tires, implementation of easy-handle model based on DFDR parameters to compute sideslip angles, application of this model to compute lateral loads on existing flight tests and incident cases, analysis of results. The last part of this work is dedicated to the preliminary study of a methodology to perform a test to retrieve lateral tire loads during ground turning with minimum requirements in terms of aircraft test instrumentation. This represents the basis for future works.

Mean curvature flow in SE (2) and applications to visual perception

Our goal in this thesis is to provide a result of existence of the degenerate non-linear, non-divergence PDE which describes the mean curvature flow in the Lie group SE(2) equipped with a sub-Riemannian metric. The research is motivated by problems of visual completion and models of the visual cortex.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Keller-Segel system, chemotaxis and applications to a mathematical model for Alzheimer's disease

In questa tesi viene presentato il modello di Keller-Segel per la chemiotassi, un sistema di tipo parabolico-ellittico che appare nella descrizione di molti fenomeni in ambito biologico e medico. Viene mostrata l'esistenza globale della soluzione debole del modello, per dati iniziali sufficientemente piccoli in dimensione N>2. La scelta di dati iniziali abbastanza grandi invece può causare il blow-up della soluzione e viene mostrato sotto quali condizioni questo si verifica. Infine il modello della chemiotassi è stato applicato per descrivere una fase della malattia di Alzheimer ed è stata effettuata un'analisi di stabilità del sistema.

Synthesis, reactivity and applications of 3,5-dimethyl-4-nitroisoxazole derivatives

3,5-dimethyl-4-nitroisoxazole derivatives are useful synthetic intermediates as the isoxazole nucleus chemically behaves as an ester, but establish better-defined interactions with chiral catalysts and lability of its N-O aromatic bond can unveil other groups such as 1,3-dicarbonyl compounds or carboxylic acids. In the present work, these features are employed in a 3,5-dimethyl-4-nitroisoxazole based synthesis of the γ-amino acid pregabalin, a medication for the treatment of epilepsy and neuropatic pain, in which this moiety is fundamental for the enantioselective formation of a chiral center by interaction with doubly-quaternized cinchona phase-transfer catalysts, whose ability of asymmetric induction will be investigated. Influence of this group in cinchona-derivatives catalysed stereoselective addition and Darzens reaction of a mono-chlorinated 3,5-dimethyl-4-nitroisoxazole and benzaldehyde will also be investigated.

Statistical mechanics of polymer models: rigorous results and applications

Monomer-dimer models are amongst the models in statistical mechanics which found application in many areas of science, ranging from biology to social sciences. This model describes a many-body system in which monoatomic and diatomic particles subject to hard-core interactions get deposited on a graph. In our work we provide an extension of this model to higher-order particles. The aim of our work is threefold: first we study the thermodynamic properties of the newly introduced model. We solve analytically some regular cases and find that, differently from the original, our extension admits phase transitions. Then we tackle the inverse problem, both from an analytical and numerical perspective. Finally we propose an application to aggregation phenomena in virtual messaging services.