In this thesis we study market risk in turbulent markets over different risk horizons. We construct portfolios which represent possible investments for a life assurance fund. The portfolios consist of equities, fixed income instruments, cash positions and interest rate derivatives. Today, the most commonly used metrics for market risk are Value-at-Risk (VaR) and Expected Shortfall (ES), and they will be central. We introduce necessary theory from quantitative finance related to asset price dynamics and security pricing. Further, interest rate related instruments are handled by the LIBOR Market Model (LMM), while equity prices are modeled as geometric Brownian motions. We use implied volatilities for instruments where they are available, and historical for the rest. We implement a risk model and make daily and quarterly market risk estimates between 2000-2008 for the portfolios. We choose some central events from the last quarter of 2008, a critical phase of the ongoing financial crisis, and analyze how the portfolios and the corresponding risk estimates are affected. Comparison of the portfolio losses against risk estimates allows us to evaluate the reliability of the broadly adopted model.

Compare the Normal Inverse Gaussian market model against empirical financial market data, and price exotic options using the numerical path integration approach.

The governing equations for waves propagating in water are derived by use of conservation laws. The equations are then cast onto dimensionless form and two important parameters are obtained. Approximations by use of asymptotic expansions in one or both of the parameters are then applied on the governing equations and we show that several different completely integrable equations, with different scaling transformations and at different order of approximations, can be derived. More precisely, the Korteweg-de Vries, Kadomtsev-Petviashvili and Boussinesq are obtained at first order, while the Camassa-Holm, Degasperis-Procesi, nonlinear Schrödinger and the Davey-Stewartson equations are obtained at second order. We discuss shortly some of the properties for each of the obtained equations.

In this text, I will look at some new approaches that may shed some light on the Kadison Singer problem, mainly one instigated by Vern Paulsen using dynamical systems in the Stone-Cech compactification of a discrete group. In order to do this, I will try to develop the theory in a crossed product setting, and look at some aspects of it that may hold interest of their own.

In this thesis an open-source software for permeability upscaling is developed. The software is based on DUNE, an open-source C++ framework for finding numerical solutions of partial differential equations (PDEs). It provides functionality used in finite elements, finite volumes and finite differences methods. Permeability is a measure of the ability of a material to transmit fluids, and determines the flow characteristics in reservoir models. Permeability upscaling is a technique to include fine-scale variations of the permeability field in a coarse-scale reservoir model. The upscaling technique used in this thesis involves solving an elliptic partial differential equation. This is solved with mixed and hybrid finite element methods. The mixed method transforms the original second order PDE into a system of two linear equations. The great advantage with these methods compared with standard finite element methods is continuity of the variable of interest in the upscaling problem. The hybrid method was introduced for being able to solve larger problems. The resulting system of equations from the hybrid method can be transformed into a symmetric positive definite system, which again can be solved with efficient iterative methods. Efficiency of the implementation is important, and as for most implementations of PDE solvers, the computational time is dominated by solving a system of linear equations. In this implementation it is used an algebraic multigrid (AMG) preconditioner provided with DUNE. This is known to be efficient on system arising from elliptic PDEs. The efficiency of the AMG preconditioner is compared with other alternatives, and is superior to the others. On the largest problem investigated, the AMG based solver is almost three times faster than the next best alternative. The performance of the implementation based on DUNE is also compared with an existing implementation by Sintef. Sintef's implementation is based on a mimetic finite difference method, but on the grid type investigated in this thesis, the methods are equivalent. Sintef's implementation uses the proprietary SAMG solver developed by Fraunhofer SCAI to solve the linear system of equations. SAMG is 58% faster than DUNE's solver on a test case consisting of 322 200 unknowns. The scalability of SAMG seem to be better than DUNE's AMG as the problem size increases. However, a great advantage with DUNE's solver is 50% lower memory usage measured on a problem consisting of approx. 3 million unknowns. Another advantage is the licensing of the software. Both DUNE and the upscaling software developed in this thesis is GPL licensed which means that anyone is free to improve or adjust the software.

We discuss nonholonomic systems in general and numerical methods for solving them. Two different approaches for obtaining numerical methods are considered; discretization of the Lagrange-d'Alembert equations on the one hand, and using the discrete Lagrange-d'Alembert principle to obtain nonholonomic integrators on the other. Among methods using the first approach, we focus on the super partitioned additive Runge-Kutta (SPARK) methods. Among nonholonomic integrators, we focus on a reversible second order method by McLachlan and Perlmutter. Through several numerical experiments the methods we present are compared by considering error-growth, conservation of energy, geometric properties of the solution and how well the constraints are satisfied. Of special interest is the comparison of the 2-stage SPARK Lobatto IIIA-B method and the nonholonomic integrator by McLachlan and Perlmutter, which both are reversible and of second order. We observe a clear connection between energy-conservation and the geometric properties of the numerical solution. To preserve energy in long-time integrations is seen to be important in order to get solutions with the correct qualitative properties. Our results indicate that the nonholonomic integrator by McLachlan and Perlmutter sometimes conserves energy better than the 2-stage SPARK Lobatto IIIA-B method. In a recent work by Jay, however, the same two methods are compared and are found to conserve energy equally well in long-time integrations.

Numerical solution of buoyancy-driven flow problems in two spatial dimensions is presented. A high-order spectral method is applied for the spatial discretization, while the temporal discretization is done by operator splitting methods. By solving the convection-diffusion equation, which governs the temperature distribution, a thorough description of both the spatial and the temporal discretization methods is given. A fast direct solver for the arising system of algebraic equations is presented, and the expected convergence rates of both the spatial and the temporal discretizations are verified. As a step towards the Navier--Stokes equations, a solution of the Stokes problem is given, where a splitting scheme technique is introduced. An extension of this framework is used to solve the incompressible Navier--Stokes equations, which govern the fluid flow. By solving the Navier-Stokes equations and the convection-diffusion equation as a coupled system, two different buoyancy-driven flow problems in two-dimensional enclosures are studied numerically. In the first problem, emphasis is put on the arising fluid flow and the corresponding thermal distribution, while the second problem mainly consists of determining critical parameters for the onset of convection rolls.

This paper investigates the solution of the free rigid body equations of motion, as well as of the equations governing the torqued rigid body. We will consider two semi-exact methods for the solution of the free rigid body equations, and we discuss the use of both rotation matrices and quaternions to describe the motion of the body; our focus is on the quaternion formulation. The approach to which we give the most attention is based on the Magnus series expansion, and we derive numerical methods of order 2, 4, 6, and 8, which are optimal as they require a minimal number of commutators. The other approach uses Gaussian quadrature to approximate an elliptic integral of the third kind. Both methods rely on the exact solution of the Euler equation which involves the exact computation of the elliptic integral of the first kind. For the solution of the torqued rigid body equations, we divide the equations into two systems where one of them is the free rigid body equations; the solutions of these two systems are then combined in the Störmer-Verlet splitting scheme. We use these methods to solve the so-called marine vessel equations. Our numerical experiments suggest that the methods we present are robust and accurate numerical integrators of both the free and the torqued rigid body.

In kernel density estimation, the most crucial step is to select a proper bandwidth (smoothing parameter). There are two conceptually different approaches to this problem: a subjective and an objective approach. In this report, we only consider the objective approach, which is based upon minimizing an error, defined by an error criterion. The most common objective bandwidth selection method is to minimize some squared error expression, but this method is not without its critics. This approach is said to not perform satisfactory in the tail(s) of the density, and to put too much weight on observations close to the mode(s) of the density. An approach which minimizes an absolute error expression, is thought to be without these drawbacks. We will provide a new explicit formula for the mean integrated absolute error. The optimal mean integrated absolute error bandwidth will be compared to the optimal mean integrated squared error bandwidth. We will argue that these two bandwidths are essentially equal. In addition, we study data-driven bandwidth selection, and we will propose a new data-driven bandwidth selector. Our new bandwidth selector has promising behavior with respect to the visual error criterion, especially in the cases of limited sample sizes.

In this master thesis we have presented a security analysis of the TLS protocol with particular emphasis on the recently discovered renegotiation attack. From our security proof we get that the Handshake protocol with renegotiation, including the fix from IETF, is secure, and hence not vulnerable to the renegotiation attack anymore. We have also analysed the Handshake protocol with session resumption, and the Application data protocol together with the Record protocol. Both of these protocols were deemed secure as well. All the security proofs are based on the UC (Universal Composability) security framework.