<p>We investigate a certain epidemics model, with and without noise. Some parameter analysis is performed together with computer simulations. The model was presented in Iacus (2008).</p>

<p>A stochastic differential equationdriven by a Brownian motion where the dispersion is determined by a parameter is considered. The parameter undergoes a change at a certain time point. Estimates of the time change point and the parameter, before and after that time, is considered.The estimates were presented in Lacus 2008. Two cases are considered: (1) the drift is known, (2) the drift is unknown and the dispersion space-independent. Applications to Dow-Jones index 1971-1974  and Goldmann-Sachs closings 2005-- May 2009 are given.</p>

<p>We apply some recent numerical solutions to diffusion bridges written in Iacus (2008). One is an approximate scheme from Bladt and S{\o}rensen (2007), another one, from  Beskos et al (2006), is an algorithm which is exact: no numerical error at given grid points!</p>

In this thesis we investigate a model of nonlinearly perturbed continuous-time renewal equation. Some characteristics of the renewal equation are assumed to have non-polynomial perturbations, more specifically they can be expanded with respect to a non-polynomial asymptotic scale. The main result of the present study is exponential asymptotic expansions for the solution of the perturbed renewal equation. These asymptotic results are also applied to various applied probability models like perturbed risk processes, perturbed M/G/1 queues and perturbed dam/storage processes. The thesis is based on five papers where the model described above is successively studied.

This paper examines the impact of several macroeconomic variables on property prices in Sweden. Linear regression is used to construct severalmathematical models relating the macroeconomic variables to property prices. Using methods of variables selection and goodness of fit measures,two final models are selected and subsequently compared, resulting in one final model. From this model, we conclude that GDP per capita, unemployment rate, inflation and repo interest rate have a significant relationship with property price changes in Sweden. Unemployment, GDPper capita, and inflation have positive relationships with property price changes, while repo interest rate has a negative relationship with propertyprice changes. However, as to what extent these variables affect property prices, no certain conclusions can be drawn from this study. / Följande studie undersöker inverkan av sex makroekonomiska variabler på bostadspriser i Sverige. Linjär regressionsanalys används för att skapaflera matematiska modeller som relaterar makroekonomiska variabler till bostadspriser. Vidare används variabelselektion och statistikor för modellevaluering för att välja ut två slutgiltiga modeller. Dessa två modeller jämförs och en slutgiltig modell väljs ut. Studiens slutsatser dras fråndenna modell. BNP per capita, arbetslöshetsgrad, inflationstakt, och reporänta har enligt den slutgiltiga modellen signifikanta förhållandentill bostadspriser i Sverige. Vidare har arbetslöshet, BNP per capita, och inflation positiva förhållanden till bostadsprisförändringar, medan reporänta har ett negativt förhållande. Studien kan inte dra några slutsatser om till vilken grad dessa variabler påverkar bostadspriser i Sverige.

This bachelor thesis within mathematical statistics studies the possibility of modelling the renewal probability for commercial non-life insurance policyholders. The project was carried out in collaboration with the non-life insurance company If P&amp;C Insurance Ltd. at their headquarters in Stockholm, Sweden. The paper includes an introduction to underlying concepts within insurance and mathematics and a detailed review of the analytical process followed by a discussion and conclusions. The first stages of the project were the initial collection and processing of explanatory insurance data and the development of a logistic regression model for policy renewal. An initial model was built and modern methods of mathematics and statistics were applied in order obtain a final model consisting of 9 significant characteristics. The regression model had a predictive power of 61%. This suggests that it to a certain degree is possible to predict the renewal probability of non-life insurance policyholders based on their characteristics. The results from the final model were ultimately translated into a measure of price sensitivity which can be implemented in both pricing models and CRM systems. We believe that price sensitivity analysis, if done correctly, is a natural step in improving the current pricing models in the insurance industry and this project provides a foundation for further research in this area. / Detta kandidatexamensarbete inom matematisk statistik undersöker möjligheten att modellera förnyelsegraden för kommersiella skadeförsärkringskunder. Arbetet utfördes i samarbete med If Skadeförsäkring vid huvudkontoret i Stockholm, Sverige. Uppsatsen innehåller en introduktion till underliggande koncept inom försäkring och matematik samt en utförlig översikt över projektets analytiska process, följt av en diskussion och slutsatser. De huvudsakliga delarna av projektet var insamling och bearbetning av förklarande försäkringsdata samt utvecklandet och tolkningen av en logistisk regressionsmodell för förnyelsegrad. En första modell byggdes och moderna metoder inom matematik och statistik utfördes för att erhålla en slutgiltig regressionsmodell uppbyggd av 9  signifikanta kundkaraktäristika. Regressionsmodellen hade en förklaringsgrad av 61% vilket pekar på att det till en viss grad är möjligt att förklara förnyelsegraden hos försäkringskunder utifrån dessa karaktäristika. Resultaten från den slutgiltiga modellen översattes slutligen till ett priskänslighetsmått vilket möjliggjorde implementering i prissättningsmodeller samt CRM-system. Vi anser att priskänslighetsanalys, om korrekt genomfört, är ett naturligt steg i utvecklingen av dagens prissättningsmodeller inom försäkringsbranschen och detta projekt lägger en grund för fortsatta studier inom detta område.

The development of scorecards for customer credit rating is a well-established field in the financial sector. The aim of this project, conducted in collaboration with a Swedish credit institute, was to develop a statistical model for predicting customer performance. In addition to conducting a model, the project also sought to identify the set of consumer characteristics with high predictive capability and how these characteristics differ when predicting performance early versus late in the loan term. To achieve this goal, a dataset containing approximately 15,000 unique loan applications approved between July 2020 and July 2022 was acquired from the credit institute, and logistic regression models were applied for different time periods ranging from 6 to 21 months. However, the models demonstrated better results than a random model but also showed difficulties in predicting creditworthiness. Possible factors contributing to the model's performance are discussed in the project, along with suggestions for potential improvements. Further research is encouraged in this area to achieve better prediction accuracy. / Utvecklingen av modeller för att bedöma kunders kreditvärdighet är en väletablerad del av finanssektorn. Som en del av ett samarbete med ett svenskt kreditinstitut var målet med detta projekt att skapa en statistisk modell som kunde predicera kunders betalningsförmåga. Utöver att skapa en modell syftar projektet också till att identifiera de egenskaper hos låntagare som har hög prediktionsförmåga samt hur dessa prediktionsvariabler skiljer sig för att förutse betalningsförmågan tidigt respektive sent in i löptiden. För att undersöka detta erhölls en datamängd innehållande cirka 15 000 unika låneansökningar som godkändes mellan juli 2020 och juli 2022 från kreditinstitutet, och logistiska regressionsmodeller tillämpades med kundernas status mellan 6 och 21 månader in av löptiden som målvariabler. Modellerna visade bättre resultat än en slumpmässig modell men visade också på stora svårigheter att förutsäga kreditvärdigheten. Möjliga faktorer som bidrar till modellernas träffssäkerhet diskuteras i projektet, tillsammans med förslag på potentiella förbättringar och ytterligare forskning uppmuntras inom detta område för att uppnå bättre modeller.

<p>The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.</p>

In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D). In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed. In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided. In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity. In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.

The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.