This thesis aims to address the Kusuoka approximation (K-approximation) within the Gatheral model using Yamada’s method, also known as the Gaussian K-approximation. Our approach begins by transforming the original Gatheral model into a model with independent Wiener processes through Cholesky decomposition. Subsequently, the system is reformulated into its Stratonovich form, facilitating the definition of vector fields and their exponentials. We will assess whether the system satisfies the Uniformly Finitely Generated (UFG) condition. Additionally, based on our calculations, a simulation code will be developed to compare our results with those obtained by Yamada.

Dependence measures are frequently applied in neuroimagining studies as a tool for analysis and classification of fMRI data. The aim of this thesis is to evaluate an algorithm for its use in classifying fMRI data using dependence measures. The focus is on evaluating the algorithm under a few changes, for example without adding voxel-based tests in voxel selection, for future use in classification. Additionally, the thesis aims to compare the performance of two dependence measures, the RV coefficient and its modified version. The classification performance of the algorithm is evaluated on a simulated fMRI data as well as resting-state and task-based fMRI data sets. On simulated fMRI data the algorithm yields an estimated accuracy of 81.41 percent versus 75.00 percent for the classifier using the RV coefficient and the modified RV coefficient, respectively. However, when evaluated on real fMRI data the estimated accuracy is close to, or even lower, than 50 percent. This indicates that the classification performance is not far from what would be expected from a classifier picked at random. It is expected that implementing additional tests to select a subset of voxels, to use in the classification step of the algorithm, may prove helpful. Further, some differences in classification performance of the RV coefficients are found. Based on the observed differences it is not possible to conclude that one measure can be preferred over the other.

Many insurance companies offer coverage for damages caused by wasps to living environments. The frequency of insurance claims related to wasps varies yearly, and anticipating the exact frequency is a complex task. By forecasting insurance claims, companies can optimize their resource allocation and management. The objective of this thesis is to forecast the frequency of insurance claims related to wasps using weather data collected over time and space by developing probabilistic models within the Bayesian framework. Weather data is used as it is assumed to capture environmental conditions that affect wasp behavior, and conditions that increase the chances of damages caused by wasps being detected. The Bayesian framework is employed as it offers an efficient way to model uncertainty by treating parameters as random. Twelve models were fitted and their predictive performance for June, July, and August were evaluated during 2022 and 2023. For June, a Negative Binomial model incorporating a spatial adjustment component with a CAR prior, and weather covariates, demonstrated the highest predictive performance. For July, a model incorporating an autoregressive parameter and the weather effect three weeks preceding the insurance claim performed best. For August, a model incorporating only weather covariates outperformed the others. The differing results show that the models capture different underlying processes in the months.

This thesis compares the efficacy of misspecified pharmacokinetic models to mechanistically true models when optimizing discrete dose levels. The aim is to compare discrete dose predictions and the resulting drug exposure levels between these models. Dose levels are discrete in practice, and physicians often determine a patient’s dose based on variables that are easily measured, such as age or weight, rather than those directly influencing pharmacokinetics, like renal function. This study investigates how using misspecified or simplified models affect the resulting drug exposure through the predicted discrete dose.Non-linear mixed effects models are used to simulate drug concentrations in hypothetical patients. Data are simulated using the true model, followed by evaluation of both true and misspecified models at the population and individual levels. At the population level, results show that simplified or misspecified models generally produced comparable exposure levels to those of the true model, despite differences in predicted doses. At the individual level, the outcomes are even more consistent, with misspecified models yielding almost identical results to their respective true models.This study underscores the practical utility of misspecified models in pharmacokinetic simulations while highlighting the importance of context-specific evaluations.

This thesis consists of five papers. In the first two papers we consider a general approach to cash flow valuation, focusing on dynamic properties of the value of a stream of cash flows. The third paper discusses immunization theory, where old results are shown to hold in general deterministic models, but often fail to be true in stochastic models. In the fourth paper we comment on the connection between arbitrage opportunities and an immunized position. Finally, in the last paper we study coherent and convex measure of risk applied to portfolio optimization and insurance.

<p>The focus of this work is on faults, and the main issue is statistical analysis and stochastic modeling of faults and fault patterns in petroleum reservoirs. The thesis consists of Part I-V and Appendix A-C. The units can be read independently. Part III is written for a geophysical audience, and the topic of this part is fault and fracture size-frequency distributions. The remaining parts are written for a statistical audience, but can also be read by people with an interest in quantitative geology. The topic of Part I and II is statistical model choice for fault size distributions, with a samling algorithm for estimating Bayes factor. Part IV describes work on spatial modeling of fault geometry, and Part V is a short note on line partitioning. Part I, II and III constitute the main part of the thesis. The appendices are conference abstracts and papers based on Part I and IV.</p> / Paper III: reprinted with kind permission of the American Geophysical Union. An edited version of this paper was published by AGU. Copyright [2000] American Geophysical Union

<p>A general time-to-event data analysis known as event history analysis is considered. The focus is on the analysis of time-to-event data using Cox's regression model when the time to the event may be measured from different origins giving several observable time scales and when longitudinal measurements are involved. For the multiple time scales problem, procedures to choose a basic time scale in Cox's regression model are proposed. The connections between piecewise constant hazards, time-dependent covariates and time-dependent strata in the dual time scales are discussed. For the longitudinal measurements problem, four methods known in the literature together with two proposed methods are compared. All quantitative comparisons are performed by means of simulations. Applications to the analysis of infant mortality, morbidity, and growth are provided.</p>

<p>This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a function of random variables. This is done with a Gaussian probability law with stochastic mean and variance. In the second paper an extension of the Generalized Hyperbolic class of probability distributions is presented. The third paper introduces, using a Gaussian probability law with stochastic mean and variance, a GARCH type stochastic process with skewed innovations. </p><p>In the fourth paper a Lévy process with second order stochastic volatility is presented, option pricing under such a process is also considered.</p>

The focus of this work is on faults, and the main issue is statistical analysis and stochastic modeling of faults and fault patterns in petroleum reservoirs. The thesis consists of Part I-V and Appendix A-C. The units can be read independently. Part III is written for a geophysical audience, and the topic of this part is fault and fracture size-frequency distributions. The remaining parts are written for a statistical audience, but can also be read by people with an interest in quantitative geology. The topic of Part I and II is statistical model choice for fault size distributions, with a samling algorithm for estimating Bayes factor. Part IV describes work on spatial modeling of fault geometry, and Part V is a short note on line partitioning. Part I, II and III constitute the main part of the thesis. The appendices are conference abstracts and papers based on Part I and IV. / Paper III: reprinted with kind permission of the American Geophysical Union. An edited version of this paper was published by AGU. Copyright [2000] American Geophysical Union

A general time-to-event data analysis known as event history analysis is considered. The focus is on the analysis of time-to-event data using Cox's regression model when the time to the event may be measured from different origins giving several observable time scales and when longitudinal measurements are involved. For the multiple time scales problem, procedures to choose a basic time scale in Cox's regression model are proposed. The connections between piecewise constant hazards, time-dependent covariates and time-dependent strata in the dual time scales are discussed. For the longitudinal measurements problem, four methods known in the literature together with two proposed methods are compared. All quantitative comparisons are performed by means of simulations. Applications to the analysis of infant mortality, morbidity, and growth are provided.