<p>First comes a description of a mathematical model of the geomagnetic field. Then some discussion of the classical non-uniqueness results of Backus. Further we look at more recent results concerning reconstruction of the geomagnetic field from intensity and the normal component of the field. New stability estimate for this reconstruction is obtained.</p>

<p>The multiple proposal methods represent a recent simulation technique for Markov Chain Monte Carlo that allows several proposals to be considered at each step of transition. Motivated by the ideas of Quasi Monte Carlo integration, we examine how strongly correlated proposals can be employed to construct Markov chains with improved mixing properties. We proceed by giving a concise introduction to the Monte Carlo and Markov Chain Monte Carlo theory, and we supply a short discussion of the standard simulation algorithms and the difficulties of efficient sampling. We then examine two multiple proposal methods suggested in the literature, and we indicate the possibility of a unified formulation of the two methods. More essentially, we report some systematic exploration strategies for the two multiple proposals methods. In particular, we present schemes for the utilization of well-distributed point sets and maximally spread search directions. We also include a simple construction procedure for the latter type of point set. A numerical examination of the multiple proposal methods are performed on two simple test problems. We find that the systematic exploration approach may provide a significant improvement of the mixing, especially when the probability mass of the target distribution is ``easy to miss'' by independent sampling. For both test problems, we find that the best results are obtained with the QMC schemes. In particular, we find that the gain is most pronounced for a relatively moderate number of proposal. With fewer proposals, the properties of the well-distributed point sets will no be that relevant. For a large number of proposals, the independent sampling approach will be more competitive, since the coverage of the local neighborhood then will be better.</p>

<p>We study and improve the CM-method for the creation of elliptic curves with specified group order over finite fields. We include a thorough review of the mathematical theory needed to understand this method. The ability to construct elliptic curves with very special group order is important in pairing-based cryptography.</p>

<p>Through a study of cored wells from the Statfjord Formation in the Tampen Area, we derive a spatially coupled classification model for facies units. We consider a Bayesian framework for the problem. A likelihood model is defined from the log-response of the formation, where response from neighbour observations are considered. A first order Markov chain prior model is estimated from the cores. From the posterior pdf, the marginal maximum posterior solution can be calculated and simulations can be generated. Since the posterior pdf can be factorized, it can be calculated by a recursive Forward-Backward algorithm for hidden Markov models. The classification model is complex, and if the model assumptions does not coincides with the underlying model, the classification might give poor results due to the large number of estimated model parameters. The results from the classification of a blind well were not as good as we expected, but gave good results for the small classes, compared to a classification model without spatial coupling.</p>

<p>This thesis explores the modeling of volatility and dependence in forward rates in the fixed income market for the purpose of risk estimation in derivative portfolios. A brief background on popular quantile-based risk measures is given. A short introduction is given to GARCH-type volatility models, as well as copula and vine models for dependence between random variables. Some details on parameter estimation and sampling related to these models are also provided. A backtesting procedure is performed using various combinations of volatility and dependence models. The results of this procedure indicate that the Student's t copula is preferable among the dependence structures considered. Also, none of the choices of conditional distribution for the volatility models provide good results at all the percentiles considered, but the normal distribution appears to be a good choice far into the tails.</p>

<p>This work will study different methods to estimate counterparty credit risk, where the methods represent both analytical approximation and simulation based method. The somewhat more analytical approximation that will be used is the current exposure method from the Bank for International Settlements and is based on simple add-on factor to the current market value. In the simulation part, Monte Carlo methods will be used. The paper will show that Monte Carlo methods enable estimation of the full exposure distribution as a function of time. From that distribution two measures of exposure will be used. The first use the peak at the 95% percentile and the second uses the concept of effective expected exposure. Those three alternative measures will be tested on six different portfolios. The portfolios are based on real data and represent both private persons, small companies, life insurance, investment bank and some of more academic interest. The estimate of exposure in those portfolios will be estimated with and without the establishment of netting agreements in order to see how that affects the exposure. The numerical results indicate that netting results in reduced exposure. In the comparisons between the different exposure measures the results show that the simulation based method in general estimates a lower exposure, but it depends intently on the construction of the portfolio. Based on those observations the main conclusion is that a simulation based approach is preferable since it enables better risk control within the firm as a consequence of enabling anatomizes of the evolution of exposure through time. Keywords: Counterparty Credit Risk, Libor Market Model and Monte Carlo simulation</p>

<p>This thesis studies the estimation of credit exposure arising from a portfolio of interest rate derivatives. The estimation is performed using a Monte Carlo simulation. The results are compared to the exposure obtained under the current exposure method provided by the Bank for International Settlements (BIS). We show that the simulation method provides a much richer set of information for credit risk managers. Also, depending on the current exposure and the nature of the transactions, the BIS method can fail to account for potential exposure. All test portfolios benefit significantly from a netting agreement, but the BIS approach tends to overestimate the risk reduction due to netting. In addition we examine the impact of antithetic variates and different time-discretizations. We find that a discretization based on derivatives' start and maturity dates may reduce simulation time significantly without loosing generality in exposure profiles. Antithetic variates have a small effect.</p>

<p>This thesis seeks to establish a methodology to reveal whether the risk appetite held by investors is dependent on the macroeconomic environment and, if present, to quantify this dependency. To do so a generic model is built and a case study is carried out with data from DnBNOR. The available data consists of the daily profit and losses together with the number and volume of transactions made in a currency portfolio owned by DnBNOR and some selected timeseries on exchange rates, all against NOK. Also, timeseries on the gross national product and consumer price index are collected from Statistics Norway. In the process of building the model, the thesis sets out the theoretical foundation for different risk measurement concepts and gives a presentation of the theory on business cycles as this is used to classify and measure the macroeconomic environment. The model is built with a Bayesian approach and implemented in WinBUGS. The use of Bayesian statistics is motivated by different time resolution of the data; some of the data is observed every day while other parts are observed each quarter. The thesis' main idea is to decompose the relevant part of the economy in one microeconomic and one macroeconomic state. The microeconomic state is unique for each day while the macroeconomic state accounts for one quarter; together they give the expected risk appetite for each day. In this way the impact from the macroeconomic state is quantified and the results show that the macroeconomic state is statistically significant for the risk appetite. As this is a case study one needs more data and research before any universal valid conclusions can be made.</p>

<p>We study the numerical integration of nonholonomic problems. The problems are formulated using Lagrangian and Hamiltonian mechanics. We review briefly the theoretical concepts used in geometric mechanics. We reconstruct two nonholonomic variational integrators from the monograph of Monforte. We also construct two one-step integrators based on a combination of the continuous Legendre transform and the discrete Legendre transform from an article by Marsden and West. Inintially these integrators display promising behavior, but they turn out to be unstable. The variational integrators are compared with a classical Runge-Kutta method. We compare the methods on three nonholonomic systems: The nonholonomic particle from the monograph of Monforte, the nonholonomic system of particles from an article by McLachlan and Perlmutter, and a variation of the Chaplygin sleigh from Bloch.</p>

<p>We present the cryptographic primitives needed in the construction of electronic voting systems based on homomorphic encryptions and on verifiable secret sharing. Then "The theory and implementation of an electronic voting system" by Ivan Damgård, Jens Groth and Gorm Salomonsen is presented as an example of electronic voting systems based on homomorphic encryptions, while "Multi-authority secret-ballot election with linear work" by Ronald Cramer, Matthew Franklin, Berry Schoenmakers and Moti Yung is presented as an example of electronic voting systems based on verifiable secret sharing. Moreover, the mathematical background for these systems are studied with particular emphasis on the security issues of the relevant sub-protocols. Comparing these two examples we find that the presented voting system based on verifiable secret sharing is more secure then the one based on homomorphic encryptions, both in regard to privacy and robustness. On the other hand, we find that the presented voting system based on homomorphic encryptions is more efficient then the one based on verifiable secret sharing.</p>