In this work, we look into the practical applicability of three joint quantile and expected shortfall regression backtests. The strict, auxiliary, and intercept ESR backtests are applied to the historical log returns of the OMX Stockholm 30 market-weight price index. We estimate the conditional variance using GARCH models for various rolling window lengths and refitting frequencies. We are particularly interested in the rejection rates of the one-sided intercept ESR backtest as it is comparable to the current standard of backtests. The one-sided test is found to perform well when the conditional variance is estimated by either the GARCH(1,1), GJR-GARCH(1,1), or EGARCH(1,1) coupled with student’s t-innovation residuals and a rolling window size of 1000 days.

The aim of this thesis is to expand on parts of the work of Björn–Björn–Shanmugalingam [2] and in particular on the construction and properties of hyperbolic fillings of nonempty bounded metric spaces. In light of [2], we introduce two new parameters λ and ξ to the construction while relaxing a specific maximal-condition. With these modifications we obtain a slightly more flexible model that generates a larger family of hyperbolic fillings. We then show that every hyperbolic filling in this family possess the desired property of being Gromov hyperbolic. Next, we uniformize an arbitrary hyperbolic filling of this type and show that, under fairly weak conditions, the boundary of the uniformization is snowflake-equivalent to the completion of the metric space it corresponds to. Finally, we show that this unifomized hyperbolic filling is a uniform space. In summary, our construction generates hyperbolic fillings which satisfy the necessary conditions for it to serve its intended purpose of an analytical tool for further studies in [2, Chapters 9-13 ] or similar. As such, it can be regarded as an improvement to the reference model.

This thesis consists of two papers, enumerated by Roman numerals. The main focus is on the spectral theory of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacians. Here, an <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacian, for integer <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />, refers to a metric graph equipped with a differential operator whose differential expression is the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?2n" />-th derivative. In Paper I, a classification of all vertex conditions corresponding to self-adjoint <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacians is given, and for these operators, a secular equation is derived. Their spectral asymptotics are analysed using the fact that the secular function is close to a trigonometric polynomial, a type of almost periodic function. The notion of the quasispectrum for <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacians is introduced, identified with the positive roots of the associated trigonometric polynomial, and is proved to be unique. New results about almost periodic functions are proved, and using these it is shown that the quasispectrum asymptotically approximates the spectrum, counting multiplicities, and results about asymptotic isospectrality are deduced. The results obtained on almost periodic functions have wider applications outside the theory of differential operators. Paper II deals more specifically with bi-Laplacians (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n=2" />), and a notion of standard conditions is introduced. Upper and lower estimates for the spectral gap --- the difference between the two lowest eigenvalues - for these standard conditions are derived. This is achieved by adapting the methods of graph surgery used for quantum graphs to fourth order differential operators. It is observed that these methods offer stronger estimates for certain classes of metric graphs. A geometric version of the Ambartsumian theorem for these operators is proved.

In this thesis we examine properties of Gelfand-Shilov spaces Ssσ and Pilipović spaces Σsσ. These are spaces of smooth functions which, along with their Fourier transforms, decay sub-exponentially. Results for the two types of spaces relating to Fourier transforms, analyticity of functions, triviality of the spaces and short-time Fourier transforms are explored. It is determined that Σsσ is nontrivial if and only if s+σ&gt;1, and that results for Ssσ when s+σ≥1 can generally be found to have corresponding counterparts for Σsσ when s+σ&gt;1.

Ever since the financial crisis in 2008 non-maturity deposits (NMDs) have had a floored deposit rate at zero. Now due to external factors some speculate that the market rate will increase. Regulations say that NMDs core deposits, which are used for further investments, must remove their rate sensitive part. In this work, high interest scenarios has been made to investigate the core deposits using an extended Vasicek model calibrated on the forward rate. Deposit rate models have been made using different regression techniques, mainly using linear models and a threshold regression model. We found that using a moving average on 21 days on Stibor 1M as a predicting variable yielded the best models. The models slope was then used to calculate the deposit rate on the given scenarioto calculate when the accounts will become rate sensitive again. At the end of the scenario, the deposits was found to decrease with 20%, 76% and 49% for the transaction-, savings account and the combined core deposits respectively using the median scenario. In order to regulate the decrease of the core deposits onecan use different rate sensitives similarly to the threshold model.

Harry Markowitz work in the 50’s spring-boarded modernportfolio theory. It gives investors quantitative tools to compose and assessasset portfolios in a systematic fashion. The main idea of the Mean-Varianceframework is that composing an optimal portfolio is equivalent to solving aquadratic optimization problem.In this project we employ the Maximally Predictable Portfolio (MPP) frameworkproposed by Lo and MacKinlay, as an alternative to Markowitz’s approach, inorder to construct investment portfolios. One of the benefits of using theformer method is that it accounts for forecasting estimation errors. Ourinvestment strategy is to buy and hold these portfolios during a time periodand assess their performance. We show that it is indeed possible to constructportfolios with high rate of return and coefficient of determination based onhistorical data. However, despite their many promising features, the success ofMPP portfolios is short lived. Based on our assessment we conclude thatinvesting in the stock market solely on the basis of the optimization resultsis not a lucrative strategy / Modern portföljteori har sitt ursprung i Harry Markowitz arbete på 50-talet. Teorin ger investerare kvantitativa verktyg för att sammansätta och utvärdera tillgångsportföljer på ett systematiskt sätt. Huvudsakligen går Markowitz idé ut på att komponera en investeringsportfölj genom att lösa ett kvadratiskt optimeringsproblem. Det här examensprojektet har utgångspunkt i Maximally Predictable Portfolio-ramverket, utvecklat av Lo och MacKinley som ett alternativ till Markowitz problemformulering, i syfte att välja ut investeringsportföljer. En av fördelarna med att använda den förra metoden är att den tar hänsyn till uppskattningsfelen från prognostisering av framtida avkastning. Vår investeringsstrategi är att köpa och behålla dessa portföljer under en tidsperiod och bedöma deras prestanda. Resultaten visar att det mha. MPP-optimering är möjligt att konstruera portföljer med hög avkastning och förklaringsvärde baserat på historisk data. Trots sina många lovande funktioner är framgången med MPP-portföljer kortlivad. Baserat på vår bedömning drar vi slutsatsen att investeringar på aktiemarknaden uteslutande på grundval av optimeringsresultatet inte är en lukrativ strategi.

We study a Cauchy problem arising in uid mechanics, involving the socalled stationary generalized Stokes system, where one should recover the ow from boundary measurements. The problem is ill-posed in the sense that the solution does not depend continuously on data. Two iterative procedures for solving this problem are proposed and investigated. These methods are regularizing and in each iteration one solves a series of well-posed problems obtained by changing the boundary conditions. The advantage with this approach, is that these methods place few restrictions on the domain and on the coefficients of the problem. Also the structure of the equation is preserved. Well-posedness of the problems used in these procedures is demonstrated, i.e., that the problems have a unique solution that depends continuously on data. Since we have numerical applications in mind, we demonstrate well-posedness for the case when boundary data is square integrable. We give convergence proofs for both of these methods.

In this Bachelor's thesis we present a few results about the absolute convergence of Fourier series, followed by an example of a differentiable function whose Fourier series does not converge absolutely. In the end we provide a suggestion for future work on generalizing the given example, and we briefly discuss an issue that has not been given much attention in the existing literature on the subject.

Centrality measures in network analysis have become a popular measurement tool for identifying coherent nodes within a network. In the context of stock markets, the centrality measure helps to identify key performing ele- ments and strengths for specific stocks and determine their impact on disrupting market value and performance. Multiple studies presented practical implementations of centrality measures for determining trends and perform- ance of a particular market. However, fewer studies applied centrality measures to predict trends in the stock market.

In calculus the concept of an infinite derivative – i.e. DF(x) = ±∞ – is seldom studied due to a plethora of complications that arise from this definition. For instance, in this extended sense, algebraic expressions involving derivatives are generally undefined; and two continuous functions possessing identical derivatives at every point of an interval generally differ by a non-constant function. These problems are fundamentally irremediable insofar as calculus is concerned and must therefore be addressed in a more general setting. This is quite difficult since the literature on infinite derivatives is rather sparse and seldom accessible to non-specialists. Therefore we supply a self-contained thesis on continuous functions with infinite derivatives aimed at graduate students with a background in real analysis and measure theory.  Predominately we study continuous primitives which satisfy the Luzin condition (N) by establishing a deep connection with the strong Luzin condition – a weak form of absolute continuity which has its origins in the Henstock–Kurzweil theory of integration. The main result states that a function satisfies the strong Luzin condition if and only if it can be expressed as a sum of two such primitives. Furthermore, we establish some pathological properties of continuous primitives which fail to satisfy the Luzin condition (N).