In this thesis we explore several statistical methods for addressing the risk of collision between two petroleum wells. Such a collision is a potentially dangerous but rare event that can occur in situations with directional drilling. In order to extend the usual approach of only considering the two closest points in the two wells in the collision risk calculations, we obtain a joint statistical distribution for the position coordinates of all the survey points in two neighboring wells.The common practice in the petroleum industry today is to use the two closest points in a hypothesis test, in order to make a conclusion on whether we should drill as planned based on the collision risk. We suggest a more accurate version of the hypothesis test, which turns out to be more conservative than the original test.As an alternative measure of the collision risk, we estimate the probability of collision. This is done in two different ways, namely by considering only the two closest points and by considering the whole wells. In the latter case, we use the joint distribution for all the survey points. For some well pair cases, the collision probability is much larger when we consider all the survey points in two wells, than when we only consider the two single closest points.We estimate the probability values by using Monte Carlo simulation methods. Since a well collision is considered to be a rare event, we introduce two methods in order to increase the accuracy in the situations where the original Monte Carlo method need an inconveniently large number of samples. These methods give accurate results even when the collision probability is very small.

A basic overview of mathematical finance and pricing theory is given. The Black-Scholes model and the LIBOR Market Model are explained, and their assumptionsare discussed and tested on historical data. The normality of log-returns of stocksand forward rates is tested for different time periods, and is found to be varyinggreatly over time. The models are calibrated using the Exponentially WeightedMoving Average (EWMA) method and implemented to perform a backtest againsthistorical data of two risk measures, Value at Risk and Expected Shortfall. Thebacktesting is done on five portfolios of varying risk, in the European and Norwegianmarkets. Three unleveraged portfolios consisting of bonds and stocks in differentproportions, and two leveraged portfolios consisting of stocks and interest rate capsrespectively are considered.The performance of the risk measures is found to be not satisfactory for all portfolios, but performance is better for riskier portfolios and assets. Variation ofperformance over different time periods is found. The periods of worst performanceare those of turbulent market conditions, notably in late 2008. These periods arefound to loosely correspond to the time periods in which log-returns of equity andforward rates are least normal.A sensitivity analysis of performances to the weighting parameter in the EWMAis done. The sensitivity is found to be substantial for all portfolios except for theportfolios holding stocks in the Norwegian market.

We have a model for simulating facies values in a rock. We can use the model to find facies structures in a 2-dimensional area, which we can use to find properties of a rock in a petroleum reservoir. The model is a Markov mesh model, with a conditional probability distribution for the facies values, with a set of parameters. By using a training image with known facies values, we can simulate the parameters in the model, and then simulate facies values for a new area. In this text, we simulate the parameters by using a Reversible jump Markov chain Monte Carlo algorithm. This lets us simulate not only the values of the parameters, but also which parameters that should be present in the model. We use the Metropolis-Hastings algorithm in the simulations. We use the model with the simulated parameters to make new images with the Markov mesh model. The images should have similar visual appearance as the training image. We are able to make images with some similar qualities as the training image, even though we are not convinced that the parameter values converge.

This master's thesis is focussed around investigating Massey products as tools for studying properties of links, in particular the Brunnian property. In the literature, there are only a few examples of the Massey product being used to study linking, none of which has any emphasis on links with the Brunnian property, except for computations for the Borromean rings. The result of the work is a number of thorough computations of Massey products in link complements, with the negative conclusion that the Massey product does not detect the Brunnian property.

Vi vil utforske bruken av adeler i matematisk fysikk.

This thesis is focused on solving the Darcy flow problem using divergence-free isogeometric methods, and comparing these results to the ones obtained using traditional finite element methods with Taylor Hood elements. A short introduction to B-splines is given, and a chapter is also about using repeated knots in the knot vectors to obtain a discontinuous basis for the finite element method. This can be useful when dealing with varying permeabilities.

In this thesis we have given an introduction to isogeometric finite element analysis on linear elasticity problems in 2D using non uniform rational B-splines (NURBS) as basis functions. We have studied the theory of B-splines and have derived the equations needed to perform linear elasticity stress analysis. An isogeometric finite element solver has been programmed in MATLAB. We have also analyzed the effect degenerated mappings have on the derivatives of the basis functions. We started by looking at a quadrilateral collapsing to a triangle, considering different parameterizations and their impact on the derivatives. We found that the derivatives were no longer in H^1 and that our basis was not a proper basis for finite element analysis. Our solution to this problem is to form a new set of basis functions by summing the basis functions at the singular points. Further we have applied this approach on a circular surface and an infinite plate with a circular hole.

We have compared the recently developed Multi-level binary replacement (MBR) design method for use in computer experiments, to the Latin hypercube design (LHD) and the Orthogonal array (OA) design. For means of comparison, we have suggested an algorithm for drawing permutations of the MBR design, so as to obtain what we have called a MBR based Latin hypercube design. In our comparison study, the main focus have been the design scores with respect to the root mean squared error (RMSE), Max and alias sum of squares criteria. We found that the MBR design generally performed good with respect to all criteria. It scored similarly to the OA design method and better than conventional Latin hypercube sampling. The score however varied with the number of samples and the set of design generators chosen for constructing the MBR design. The MBR design performed better for designs with a relatively high number of samples compared to the number of factors.

We prove that quasicrystals are universal sets of stable sampling in any dimension. Necessary and sufficient density conditions for stable sampling and interpolation sets in one dimension are studied in detail.

Using a simulation based model, with the Black-Scholes framework for equity andThe LIBOR Market Model for interest rates, we study market risk in multi assetclassportfolios, with static and dynamic weighting. The risk measures consideredare Value-at-Risk and Expected-Tail-Loss. The theoretical foundation is introducedand imperfections in the models and their assumptions are pointed out.The validity of the models and risk measures is tested using a backtesting procedureagainst data ranging from September 1999 to September 2009, with particularemphasis on the turbulent period of 2007 to September 2009. The results indicatethat the models perform slightly worse on the portfolio with the added complexityof a dynamic weighting regime. No evidence of the models performing lesssatisfactory under the latest financial turbulence is found.