Random field models have provided a flexible environment in which the properties of the term structure of interest rates are captured almost as observed. In this study we provide an overview of the forward rate random fiield models and propose an extension in which the forward rates fluctuate along with a two parameter process represented by a random field. We then provide a mathematical expression of the yield curve under this model and sketch the prospective utilities and applications of this model for interest rate management.

Life insurance companies rely on mortality rate models to set appropriate premiums for their services. Over the past century, average life expectancy has increased and continues to do so, necessitating more accurate models. Two commonly used models are the Gompertz-Makeham law of mortality and the Lee-Carter model. The Gompertz-Makeham model depends solely on an age variable, while the Lee-Carter model incorporates a time-varying aspect which accounts for the increase in life expectancy over time. This paper constructs both models using training data acquired from Skandia Mutual Life Insurance Company and compares them to validation data from the same set. The study suggests that the Lee-Carter model may be able to offer some improvements compared to the Gompertz-Makeham law of mortality in terms of predicting future mortality rates. However, due to a lack of qualitative data, creating a competitive Lee-Carter model through Singular Value Decomposition, SVD, proved to be problematic. Switching from the current Gompertz-Makeham model to the Lee-Carter model should, therefore, be explored further when more high quality data becomes available.

Existing traffic models are widely used in multiple frameworks, most prominently, microscopic vehicle-scale occurring on the scale of seconds and macroscopic city-scale flow patterns that develop over the scale of hours. Research works and practical applications usually employ either one or the other framework, and there is little overlap in the respective research communities. This dissertation develops mathematical techniques to bridge the two scales. The particular case of bridging the micro and macro scales of models in the stable traffic regime has been extensively studied, however what has been often overlooked is the unstable regime. Thus, of particular importance are models that can capture dynamic instabilities and traveling traffic waves called phantom jams. Such models are particularly challenging to analyze, as many papers on PDE models explicitly exclude the unstable situation. This thesis (i) outlines the mathematical foundations of microscopic and macroscopic models of interest, (ii) establishes a principled procedure of generating macroscopic flow quantities from microscopic models in the unstable regime, (iii) presents a study addressing the averaging of scales and the understanding of macroscopic manifestations of microscopic car-following traffic waves based on a framework of systematic hierarchy of tests that isolate the car-following dynamics, (iv) explains the corresponding effective traffic state and non-equilibrium wave structures that rise in the fundamental diagram, (v) and derives and validates vehicle type specific simple fuel consumption rate models that are accurate, computationally fast, and have desirable physics-like properties. The insights gained from this study has many applications. One of them presented here is the relevance of dampening traffic waves in the presence of sparse control and in light of the energy demand of traffic at the vehicle-scale, waves-scale, and city scale. / Mathematics

Title: Stochastic interest rates modeling Author: Jakub Černý Abstract: This present work studies different stochastic models of interest rates. Theoretical part of this work describes short-rate models, HJM fra- mework and LIBOR Market model. It focuses in detail on widely known short-rate models, i.e. Vašíček, Hull-White and Ho-Lee model, and on LI- BOR Market model. This part ends by valuation of interest rate options and model calibration to real data. Analytical part of the work analyses valuation of real non-standard interest rate derivative using different models. Part of this derivative valuation is comparison among models in terms of general valuation and also in terms of capturing the dynamics of interest rates. The aim of this work is to describe different stochastic models of interest rates and mainly to compare them with each other.

The scope of this study is to estimate the zero-coupon yield curve of tomorrow by using Vasicek yield curve model with the zero-coupon bond yield data of today. The raw data of this study is the yearly simple spot rates of the Turkish zero-coupon bonds with different maturities of each day from July 1, 1999 to March 17, 2004. We completed the missing data by using Nelson-Siegel yield curve model and we estimated tomorrow yield cuve with the discretized Vasicek yield curve model.

Dans cette thèse, nous étudions le vaste domaine des neurosciences à travers des outils théoriques, numériques et expérimentaux. Nous étudions comment les modèles à taux de décharge peuvent être utilisés pour capturer différents phénomènes observés dans le cerveau. Nous étudions les régimes dynamiques des réseaux couplés de neurones excitateurs (E) et inhibiteurs (I): Nous utilisons une description fournie par un modèle à taux de décharge et la comparons avec les simulations numériques des réseaux de neurones à potentiel d'action décrits par le modèle EIF. Nous nous concentrons sur le régime où le réseau EI présente des oscillations, puis nous couplons deux de ces réseaux oscillants pour étudier la dynamique résultante. La description des différents régimes pour le cas de deux populations est utile pour comprendre la synchronisation d'une chaine de modules E-I et la propagation d'ondes observées dans le cerveau. Nous examinons également les modèles à taux de décharge pour décrire l'adaptation sensorielle: Nous proposons un modèle de ce type pour décrire l'illusion du mouvement consécutif («motion after effect», (MAE)) dans la larve du poisson zèbre. Nous comparons le modèle à taux de décharge avec des données neuronales et comportementales nouvelles. / In this thesis, we investigate the vast field of neuroscience through theoretical, numerical and experimental tools. We study how rate models can be used to capture various phenomena observed in the brain. We study the dynamical regimes of coupled networks of excitatory (E) and inhibitory neurons (I) using a rate model description and compare with numerical simulations of networks of neurons described by the EIF model. We focus on the regime where the EI network exhibits oscillations and then couple two of these oscillating networks to study the resulting dynamics. The description of the different regimes for the case of two populations is helpful to understand the synchronization of a chain of E-I modules and propagation of waves observed in the brain. We also look at rate models of sensory adaptation. We propose one such model to describe the illusion of motion after effect in the zebrafish larva. We compare this rate model with newly obtained behavioural and neuronal data in the zebrafish larva.

When modeling interest rates, using strong formulations of underlying differential equations is prone to bad numerical approximations and high computational costs, due to close to non-smoothness in the probability density function of the interest rate. To circumvent these problems, a weak formulation of the Fokker–Planck equation using Gaussian radial basis functions is suggested. This approach is used in a parameter estimation process for two interest rate models: the Vasicek model and the Cox–Ingersoll–Ross model. In this thesis, such an approach is shown to yield good numerical approximations at low computational costs.

We estimate a multivariate unobserved components stochastic volatility model to explain the dynamics of a panel of six exchange rates against the US Dollar. The empirical model is based on the assumption that both countries' monetary policy strategies may be well described by Taylor rules with a time-varying inflation target, a time-varying natural rate of unemployment, and interest rate smoothing. The estimates closely track major movements along with important time series properties of real and nominal exchange rates across all currencies considered. The model generally outperforms a benchmark model that does not account for changes in trend inflation and trend unemployment. (authors' abstract) / Series: Department of Economics Working Paper Series

This diploma thesis deals with short-term interest rate models. Many interest models have been developed in the last decades. They focus on accuracy of prediction. The pioneering one was developed by Vasicek in 1977 followed by the work of others. Nowadays these vary in their level of comprehensiveness and technical difficulty. The main aim of the thesis is to introduce not only a basic Vasicek's work but also more sophisticated models such as Brennan-Schwartz or Longstaff-Schwartz.

An in depth study of the pricing of Game contingent claims under a general diffusion market model, in which interest rate is non constant, is presented. With the idea of providing a few numerical examples of the valuation of such claims, we present a detailed description of a Bootstrapping procedure to obtain interest rate information from Swaps rates. We also present a Stripping procedure that can be used to obtain initial spot (caplet) volatility from Market quotes on Caps/FLoors. These methods are of general application and could be used in the calibration of diffusion models of interest rate. Then we show several examples of calibration of the Hull--White model of interest rates. Our calibration examples are later used in the numerical approximation of the value of a particular form of Game option.