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.

Stochastic local volatility model for fx markets

Questa tesi verte sullo studio di un modello a volatilità stocastica e locale, utilizzato per valutare opzioni esotiche nei mercati dei cambio. La difficoltà nell'implementare un modello di tal tipo risiede nella calibrazione della leverage surface e uno degli scopi principali di questo lavoro è quello di mostrarne la procedura.

A mathematical model of the early visual system and border completion via subriemannian Hamiltonian geodesics

In this thesis, we aim to discuss a simple mathematical model for the edge detection mechanism and the boundary completion problem in the human brain in a differential geometry framework. We describe the columnar structure of the primary visual cortex as the fiber bundle R2 × S1, the orientation bundle, and by introducing a first vector field on it, explain the edge detection process. Edges are detected through a lift from the domain in R2 into the manifold R2 × S1 and are horizontal to a completely non-integrable distribution. Therefore, we can construct a subriemannian structure on the manifold R2 × S1, through which we retrieve perceived smooth contours as subriemannian geodesics, solutions to Hamilton’s equations. To do so, in the first chapter, we illustrate the functioning of the most fundamental structures of the early visual system in the brain, from the retina to the primary visual cortex. We proceed with introducing the necessary concepts of differential and subriemannian geometry in chapters two and three. We finally implement our model in chapter four, where we conclude, comparing our results with the experimental findings of Heyes, Fields, and Hess on the existence of an association field.

Local and nonlocal integro-differential reaction-diffusion models for protein dynamics in Alzheimer's disease

In this work, integro-differential reaction-diffusion models are presented for the description of the temporal and spatial evolution of the concentrations of Abeta and tau proteins involved in Alzheimer's disease. Initially, a local model is analysed: this is obtained by coupling with an interaction term two heterodimer models, modified by adding diffusion and Holling functional terms of the second type. We then move on to the presentation of three nonlocal models, which differ according to the type of the growth (exponential, logistic or Gompertzian) considered for healthy proteins. In these models integral terms are introduced to consider the interaction between proteins that are located at different spatial points possibly far apart. For each of the models introduced, the determination of equilibrium points with their stability and a study of the clearance inequalities are carried out. In addition, since the integrals introduced imply a spatial nonlocality in the models exhibited, some general features of nonlocal models are presented. Afterwards, with the aim of developing simulations, it is decided to transfer the nonlocal models to a brain graph called connectome. Therefore, after setting out the construction of such a graph, we move on to the description of Laplacian and convolution operations on a graph. Taking advantage of all these elements, we finally move on to the translation of the continuous models described above into discrete models on the connectome. To conclude, the results of some simulations concerning the discrete models just derived are presented.

Numerical study on viscoelastic behavior of polypropylene fiber reinforced concrete subjected to temperature variations using LDPM theory

This thesis is focused on the viscoelastic behavior of macro-synthetic fiber-reinforced concrete (MSFRC) with polypropylene studied numerically when subjected to temperature variations (-30 oC to +60 oC). LDPM (lattice discrete particle model), a meso-scale model for heterogeneous composites, is used. To reproduce the MSFRC structural behavior, an extended version of LDPM that includes fiber effects through fiber-concrete interface micromechanics, called LDPM-F, is applied. Model calibration is performed based on three-point bending, cube, and cylinder test for plain concrete and MSFRC. This is followed by a comprehensive literature study on the variation of mechanical properties with temperature for individual fibers and plain concrete. This literature study and past experimental test results constitute inputs for final numerical simulations. The numerical response of MSFRC three-point bending test is replicated and compared with the previously conducted experimental test results; finally, the conclusions were drawn. LDPM numerical model is successfully calibrated using experimental responses on plain concrete. Fiber-concrete interface micro-mechanical parameters are subsequently fixed and LDPM-F models are calibrated based on MSFRC three-point bending test at room temperature. Number of fibers contributing crack bridging mechanism is computed and found to be in good agreement with experimental counts. Temperature variations model for individual constituents of MSFRC, fibers and plain concrete, are implemented in LDPM-F. The model is validated for MSFRC three-point bending stress-CMOD (crack mouth opening) response reproduced at -30 oC, -15 oC, 0 oC, +20 oC, +40 oC and +60 oC. It is found that the model can well describe the temperature variation behavior of MSFRC. At positive temperatures, simulated responses are in good agreement. Slight disagreement in negative regimes suggests an in-depth study on fiber-matrix interface bond behavior with varying temperatures.