Several manufacturers of road vehicles today are working on developing autonomous vehicles. One subject that is often up for discussion when it comes to integrating autonomous road vehicles into the infrastructure is the safety aspect. There is in the context no common view of how safety should be quantified. As a contribution to this discussion we propose describing each potential hazardous event of a vehicle as a Semi-Markov Process (SMP). A reliability-based method for using the semi-Markov representation to calculate the probability of a hazardous event to occur is presented. The method simplifies the expression for the reliability using the Laplace-Stieltjes transform and calculates the transform of the reliability exactly. Numerical inversion algorithms are then applied to approximate the reliability up to a desired error tolerance. The method is validated using alternative techniques and is thereafter applied to a system for automated steering based on a real example from the industry. A desired evolution of the method is to involve a framework for how to represent each hazardous event as a SMP. / Flertalet tillverkare av vägfordon jobbar idag på att utveckla autonoma fordon. Ett ämne ofta på agendan i diskussionen om att integrera autonoma fordon på vägarna är säkerhet. Det finns i sammanhanget ingen klar bild över hur säkerhet ska kvantifieras. Som ett bidrag till denna diskussion föreslås här att beskriva varje potentiellt farlig situation av ett fordon som en Semi-Markov process (SMP). En metod presenteras för att via beräkning av funktionssäkerheten nyttja semi-Markov representationen för att beräkna sannolikheten för att en farlig situation ska uppstå. Metoden nyttjar Laplace-Stieltjes transformen för att förenkla uttrycket för funktionssäkerheten och beräknar transformen av funktionssäkerheten exakt. Numeriska algoritmer för den inversa transformen appliceras sedan för att beräkna funktionssäkerheten upp till en viss feltolerans. Metoden valideras genom alternativa tekniker och appliceras sedan på ett system för autonom styrning baserat på ett riktigt exempel från industrin. En fördelaktig utveckling av metoden som presenteras här skulle vara att involvera ett ramverk för hur varje potentiellt farlig situation ska representeras som en SMP.

This thesis considers new methods that exploit recent developments in computer technology to address three extant problems in the area of Finance and Econometrics. The problem of Asian option pricing has endured for the last two decades in spite of many attempts to find a robust solution across all parameter values. All recently proposed methods are shown to fail when computations are conducted using standard machine precision because as more and more accuracy is forced upon the problem, round-off error begins to propagate. Using recent methods from numerical analysis based on multi-precision arithmetic, we show using the Mathematica platform that all extant methods have efficacy when computations use sufficient arithmetic precision. This creates the proper framework to compare and contrast the methods based on criteria such as computational speed for a given accuracy. Numerical methods based on a deformation of the Bromwich contour in the Geman-Yor Laplace transform are found to perform best provided the normalized strike price is above a given threshold; otherwise methods based on Euler approximation are preferred. The same methods are applied in two other contexts: the simulation of stable distributions and the computation of unit root densities in Econometrics. The stable densities are all nested in a general function called a Fox H function. The same computational difficulties as above apply when using only double-precision arithmetic but are again solved using higher arithmetic precision. We also consider simulating the densities of infinitely divisible distributions associated with hyperbolic functions. Finally, our methods are applied to unit root densities. Focusing on the two fundamental densities, we show our methods perform favorably against the extant methods of Monte Carlo simulation, the Imhof algorithm and some analytical expressions derived principally by Abadir. Using Mathematica, the main two-dimensional Laplace transform in this context is reduced to a one-dimensional problem.