Mimicking the one-dimensional marginal distributions of stochastic diffusions

In the large maturity limit, we compute explicitly the Local Volatility surface for Heston, through Dupire’s formula, with Fourier pricing of the respective derivatives of the call price. Than we verify that the prices of European call options produced by the Heston model, concide with those given by the local volatility model where the Local Volatility is computed as said above.

One dimensional extended Hubbard model: two-particle bound states and resonances.

In questo lavoro di tesi è stato svolto uno studio analitico sul modello di Hubbard esteso unidimensionale al fine di osservare la presenza di eventuali risonanze che possano dare origine alla formazione di stati legati di due particelle. L'esistenza di uno stato legato stabile ha suscitato grande interesse negli ultimi anni, sia in ambito teorico che sperimentale, poichè è alla base di molti fenomeni che vengono osservati nei sistemi a molti corpi a basse temperature, come il BCS-BEC crossover. Pertanto si è ritenuto utile studiare il problema a due corpi nel modello di Hubbard esteso, che in generale non è integrabile. Il modello considerato contiene interazioni a primi e secondi vicini, in aggiunta all'interazione di contatto presente nel modello di Hubbard. Il problema è stato indagato analiticamente attraverso il Bethe ansatz, che consente di trovare tutti gli autovalori e le autofunzioni dell'Hamiltoniana. L'ansatz di Bethe sulla funzione d'onda è stato generalizzato per poter tener conto dei termini di interazione a più lungo raggio rispetto all'interazione di contatto. Si trova che, in questo modello, nel limite termodinamico, possono avvenire delle risonanze (o quasi-risonanze) in cui la lunghezza di scattering diverge, contrariamente a quanto avviene nel modello di Hubbard. Tale fenomeno si verifica quando il livello energetico discreto degli stati legati “tocca” la banda di scattering. Inoltre, con l'aggiunta di nuovi termini di interazione emergono nuovi stati legati. Nel caso in esame, si osservano due famiglie di stati legati, se lo spin totale delle due particelle è 1, e tre famiglie di stati legati, se lo spin totale è 0.

Organic matter deposition/resuspension in a one-dimensional physical-biogeochemical model. A modelling study

The shallow water configuration of the gulf of Trieste allows the propagation of the stress due to wind and waves along the whole water column down to the bottom. When the stress overcomes a particular threshold it produces resuspension processes of the benthic detritus. The benthic sediments in the North Adriatic are rich of organic matter, transported here by many rivers. This biological active particulate, when remaining in the water, can be transported in all the Adriatic basin by the basin-wide circulation. In this work is presented a first implementation of a resuspension/deposition submodel in the oceanographic coupled physical-biogeochemical 1-dimensional numerical model POM-BFM. At first has been considered the only climatological wind stress forcing, next has been introduced, on the surface, an annual cycle of wave motion and finally have been imposed some exceptional wave event in different periods of the year. The results show a strong relationship between the efficiency of the resuspension process and the stratification of the water column. During summer the strong stratification can contained a great quantity of suspended matter near to the bottom, while during winter even a low concentration of particulate can reach the surface and remains into the water for several months without settling and influencing the biogeochemical system. Looking at the biologic effects, the organic particulate, injected in the water column, allow a sudden growth of the pelagic bacteria which competes with the phytoplankton for nutrients strongly inhibiting its growth. This happen especially during summer when the suspended benthic detritus concentration is greater.

Dispersion and energy relations in periodic chains and grids

In this study wave propagation, dispersion relations, and energy relations for linear elastic periodic systems are analyzed. In particular, the dispersion relations for monoatomic chain of infinite dimension are obtained analytically by writing the Block-type wave equation for a unit cell in order to capture the dynamic behavior for chains under prescribed vibration. By comparing the discretized model (mass-spring chain) with the solid bar system, the nonlinearity of the dispersion relation for chain indicates that the periodic lattice is dispersive in contrast to the continuous rod, which is non dispersive. Further investigations have been performed considering one-dimensional diatomic linear elastic mass-spring chain. The dispersion relations, energy velocity, and group velocity have been derived. At certain range of frequencies harmonic plane waves do not propagate in contrast with monoatomic chain. Also, since the diatomic chain considered is a linear elastic chain, both of the energy velocity and the group velocity are identical. As long as the linear elastic condition is considered the results show zero flux condition without residual energy. In addition, this paper shows that the diatomic chain dispersion relations are independent on the unit cell scheme. Finally, an extension for the study covers the dispersion and energy relations for 2D- grid system. The 2x2 grid system show a periodicity of the dispersion surface in the wavenumber domain. In addition, the symmetry of the surface can be exploited to identify an Irreducible Brillouin Zone (IBZ). Compact representations of the dispersion properties of multidimensional periodic systems are obtained by plotting frequency as the wave vector’s components vary along the boundary of the IBZ, which leads to a widely accepted and effective visualization of bandgaps and overall dispersion properties.