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In this thesis, which consists of four papers, approximation of optimal control problems is studied. In Paper I the Symplectic Pontryagin method for approximation of optimally controlled ordinary differential equations is presented. The method consists of a Symplectic Euler time stepping scheme for a Hamiltonian system with a regularized Hamiltonian. Under some assumptions it is shown that the approximate value function associated with this scheme converges to the original value function with a linear rate. In Paper II the ideas from Paper I are extended to approximation of an optimally controlled partial differential equation, a one-dimensional Ginzburg-Landau equation. The approximation is performed in two steps. In the first step a value function associated with a finite element spatial discretization is shown to converge quadratically in the mesh size to the original value function. In the second step a Symplectic Euler discretization in time is shown to converge with a linear rate. The behavior of optimal solutions is shown by numerical examples. In Paper III the same approximation method as in Paper II is applied to three other problems; the optimal design of an electric conductor, the design of an elastic domain, and the problem of reconstructing the interior of an object from measured electrical surface currents. Since these problems are time-independent the Hamilton-Jacobi theory can not be used. In order to be able to obtain error bounds the problems are therefore transferred to a setting where time plays a role. Computer experiments with the Symplectic Pontryagin method is performed for all three problems. Common to the three first papers is that the convergence proofs use that the approximate value functions solve Hamilton-Jacobi equations consistent with the original Hamilton-Jacobi equations. Paper IV concerns convergence of attainable sets for non-convex differential inclusions. When the right hand side in the differential inclusion is a bounded, Lipschitz set-valued function it is shown that the convergence in Hausdorff-distance of attainable sets for a Forward Euler discretization is linear in the time step. This implies that dynamic programming using Forward Euler discretizations of optimal control problems converge with a linear rate when all the functions involved are bounded and Lipschitz continuous. / QC 20100917

Diss. Linköping : Linköpings universitet, 2006.

Diss. Stockholm : Stockholms universitet, 2006.

Diss. (sammanfattning) Umeå : Umeå universitet, 2006.

Risk management is practiced in many financial institutions and one of the most commonly used risk measures is Value at Risk. This measure represents how much a portfolio of assets could lose over a pre-specified time horizon to a cer- tain probability. Value at Risk is often utilized to calculate capital requirements and margins, which work as collateral to cover potential losses that might occur due to market turbulence. It is important that the calculation of Value at Risk is accurate which require complex and time demanding models but many financial institutions also wishes to calculate Value at Risk continuously throughout the day, which requires computational speed. Today’s most commonly used method for calculating Value at Risk is his- torical simulation which is a simple but often inaccurate method. It is criticized by many scholars since it heavily depends on the assumption that history will repeat itself. A substitute method to historical simulation is the Monte Carlo simulation which is seen as a more accurate and robust method. However, for a high-dimensional portfolio, Monte Carlo simulated Value at Risk is very com- putationally demanding and in many cases it is not possible to use it due to time constraints. The study investigates alternative methods for calculating Value at Risk with the purpose of finding a method that could be used as a substitute to the Monte Carlo method. The portfolio used in this thesis is a high-dimensional equity portfolio containing 2520 equities with 10 years of observations. I find that by first using a clustering algorithm to divide the equities in to groups based on their correlation, and then applying principal component analysis to achieve a lower dimensional problem, computational time can be reduced by approxi- mately 99% and still provide an accurate result.

Syftet med denna studie var att undersöka vilka färdigheter och strategier elever i grundskolans årskurs 2 och 3 i fyra klasser visar vid avläsning av den analoga och den digitala klockan. Den teoretiska utgångspunkten var variationsteorin. För att skaffa oss kunskap om vilka färdigheter de sammanlagt 86 eleverna besitter användes ett diagnosmaterial som kompletterades med intervjuer för att utröna elevernas strategier. Dessa intervjuer gjordes med två elever från varje klass där en hade ett högre resultat än klassens medelvärde och den andra hade ett lägre. Resultatet visade att de färdigheter eleverna visade i diagnosen var varierande, från att endast kunna den analoga klockans hel- och halvtimma till att obehindrat kunna röra sig mellan samtliga analoga och digitala tidpunkter. Under elevintervjuerna framkom det att strategin 5-minuterssteg användes frekvent av flera elever. Eleverna kunde använda flera strategier samtidigt, däribland 60-kompisarna, milstolpar, mentala mönster samt att använda sig av konkret material. Det framfördes också att tidsbegrepp upplevdes som svåra, därför användes strategier för att förenkla dem. Även den felaktiga strategin att avläsa timvisaren som minutvisare omnämndes.

This thesis aims to investigate the possibility if hidden Markov models (HMM) can be used for inference about operational risk given financial time series data of Auditchanges and Audit prices. The models tested vary in the number of possible states each underlying latent process can take. All models have been implemented usingR-statisticalsoftware along with the depmixS4 package. From the evaluation of the work, it was shown that there was a clear difference between the states, according to the the types of observation they emitted, for the final model. The thesis shows that the biggest factors affecting operational risk were the number of changes of the trades and the time between those changes. It also showed that it was, in large part, the same trader who carried out all the trades as well as changes and only within the internal department. The final conclusion is therefore that HMMs are possible and appropriate to use for inference about operational risk, but that more labeled data are required to express the models predictive performance.