In this licentiate thesis we study some mathematical problems in hydrodynamic lubrication theory. It is composed of two papers (A and B) and a complementary appendix. Lubrication theory is devoted to fluid flow in thin domains. The main purpose of lubrication is to reduce friction and wear between two solid surfaces in relative motion. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation which describes the motion of viscous fluids. In thin domains several approximations are possible which leads to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict different important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field.In many practical situations the surface roughness amplitude and the film thickness are of the same order. Therefore, any realistic model should account for the effect of surface roughness. This implies that the mathematical modelling leads to partial differential equations with coefficients that will oscillate rapidly in space and time due to the relative motion of the surfaces. A direct numerical analysis is very difficult since an extremely fine mesh is required to describe the different scales. One method which has proved successful to handle such problems is to do some averaging (asymptotic analysis). The branch in mathematics which has been developed for this purpose is called homogenization.In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness ε and wavelength μ of the roughness tend to zero is analyzed and described. The results are obtained using the formal method of multiple scale expansion. The limit equation depends on how fast the two small parameters ε and μ go to zero relative to each other. Three different limit equations are derived. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high frequency roughness regime).In paper B we present a mathematical model in hydrodynamic lubrication that takes into account cavitation (formation of air bubbles), surface roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness. A one-dimensional problem describing a step bearing is also solved explicitly and by numerical methods.

This Licentiate thesis deals with the theory of Hardy-type inequalities in anew situation, namely when the classical Hardy operator is replaced by amore general operator with kernel. The kernels we consider belong to thenew classes O+ n and O-n , n = 0; 1; :::, which are wider than co-called Oinarovclass of kernels.The thesis consists of three papers (papers A, B and C), an appendix topaper A and an introduction, which gives an overview to this specific fieldof functional analysis and also serves to put the papers in this thesis into amore general frame.In paper A some new Hardy-type inequalities for the case with Hardy-Volterra integral operators involved are proved and discussed. The case 1

The present thesis is devoted to derivation of Darcy’s Law for incompressible Newtonian fluid in perforated domains by means of homogenization techniques.The problem of describing flow in porous media occurs in the study of various physical phenomena such as filtration in sandy soils, blood circulation in capillaries etc. In all such cases physical quantities (e.g. velocity, pressure) are dependent of the characteristic size ε 1 of the microstructure of the fluid domain. However in most practical applications the significant role is played by averaged characteristics, such as permeability, average velocity etc., which do not depend on the microstructure of the domain. In order to obtain such quantities there exist several mathematical techniques collectively referred to as homogenization theory.This thesis consists of two papers (A and B) and complementary appendices. We assume that the flow is governed by the Stokes equation and that global normal stress boundary condition and local no-slip boundary condition are satisfied. Such mixed boundary condition is natural for many applications and here we develop the rigorous mathematical theory connected to it. The assumption of mixed boundary condition affects on corresponding forms of Darcy’s law in both papers and raises some essential difficulties in analysis in Paper A.In both papers the perforated domain is supposed to have periodical structure and the fluid to be incompressible and Newtonian. In Paper A the situation described above is considered in a framework of rigorous functional analysis, more precisely the theorem concerning the existence and uniqueness of weak solutions for the Stokes equation is proved and Darcy’s law is obtained by using two-scale convergence procedure. As it was mentioned, vast part of this paper is devoted to adaptation of classical results of functional analysis to the case of mixed boundary condition.In Paper B the Navier–Stokes system with mixed boundary condition is studied in thin perforated domain. In such cases it is natural to introduce another small parameter δ which corresponds to the thickness of the domain (in addition to the perforation parameter ε). For the case of thin porous medium the asymptotic behavior as both the film thickness δ and the perforation period ε tend to zero at different rates is investigated. The results are obtained by using the formal method of asymptotic expansions. Depending on how fast the two small parameters δ and ε go to zero relative to each other, different forms of Darcy’s law are obtained in all three limit cases — very thin porous medium (δ ε), proportionallythin porous medium (δ ∼ λε, λ ∈ (0,∞)) and homogeneously thin porous medium (δ ε).

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This Licentiate thesis is devoted to the study of mapping properties of different operators (Hardy type, singular and potential) between various function spaces. The main body of the thesis consists of five papers and an introduction, which puts these papers into a more general frame. In paper A we prove the boundedness of the Riesz Fractional Integration Operator from a Generalized Morrey Space to a certain Orlicz-Morrey Space, which covers the Adams resultfor Morrey Spaces. We also give a generalization to the case of Weighted Riesz Fractional Integration Operators for a class of weights. In paper B we study the boundedness of the Cauchy Singular Integral Operator on curves in complex plane in Generalized Morrey Spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of Singular Integral Operators in Weighted Generalized Morrey Spaces. In paper C we prove the boundedness of the Potential Operator in Weighted Generalized Morrey Spaces in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation with a free term in such a space. We also give a short overview of some typical situations when Potential type Operators arise when solving PDEs. ​In paper D some new inequalities of Hardy type are proved. More exactly, the boundedness of multidimensional Weighted Hardy Operators in Hölder Spaces are proved in cases with and without compactification. In paper E the mapping properties are studied for Hardy type and Generalized Potential type Operators in Weighted Morrey type Spaces.

<p>Examensarbetet är utfört vid Institutionen för teknik och naturvetenskap (ITN) vid Tekniska fakulteten, Linköpings universitet</p>

The purpose of this essay is to prove the existence of the Riemann-Stieltjes integral. After doing so, we present some applications in complex analysis, where we define the complex curve integral as a special case of the Riemann- Stieltjes integral, and then focus on Cauchy’s celebrated integral theorem. To show the versatility of the Riemann-Stieltjes integral, we also present some applications in probability theory, where the integral generates a general formula for the expectation, regardless of its underlying distribution.

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Since the new risk sensitive regulation Solvency II was enabled the 1st of January 2016 the European insurance companies have to review their investment strategies. Insurance companies are among the largest institutional investors in Europe holding EUR 6.7 trillion assets, thus major changes in their asset management can impact the capital markets. To investigate how the investing opportunities have changed for life insurance companies, a representative Swedish life insurance company with an occupational pension portfolio was simulated for thirty years. This was made by first simulating the money market, bonds, equities and real estate for the simulated time by a stochastic multivariate process. Using Modern Portfolio Theory the portfolio weights was constructed for the financial asset portfolios for the model of the company. To determine future liabilities a representative ITP 2 pension portfolio was modelled where the pension policies was priced using traditional life insurance pricing theory in continuous time. For the company to be representative actuarial assumptions and as well as a consolidation policy was constructed in line with the major traditional life insurance companies in Sweden. The simulations of the company resulted in monthly cash flows, development of life insurance mathematical functions and the solvency capital requirements. The solvency capital requirement by Solvency II was calculated by applying the standard formula handed by EIOPA, where for life insurance companies the market risk module dominates in contribution to the capital requirement. By comparing the new risk sensitive capital requirement with the solvency capital requirement by the old regulations a change of structure dependent on time and asset allocation was observed. The Solvency II capital requirement for life insurance companies is clearly more dependent on the financial asset strategy for the company whereas the old capital requirement is not. The structure of the new capital requirement follows the same structure as the solvency market risk module where it is clear that low risk portfolios does not necessarily correspond to a lower capital requirement. The conclusion of this thesis is that life insurance companies in Sweden have tightened financial investing opportunities. This is due to Solvency II since this regulation is more risk sensitive than the old regulation. / Sedan det nya riskkänsliga regelverket Solvens II trädde i kraft den första januari 2016 behöver europeiska försäkringsbolag se över sin investeringsstrategi för finansiella tillgångar. Försäkringsbolag är bland de största finansiella investeringsinstituten i Europa med ett innehav om 6,7 biljoner euro och i och med detta kan stora förändringar i försäkringsbolagens tillgångsallokering påverka kapitalmarknaden. För att undersöka hur investeringsmöjligheterna har förändrats för livförsäkringsbolag simulerades ett svenskt fiktivt och representativt livförsäkringsbolag med en tjänstepensionsportfölj trettio år framåt i tiden. Först simulerades penningmarknaden, obligationer, aktier och fastighetsmarknaden trettio år med en multivariat stokastisk process. Genom att tillämpa modern portföljteori konstruerades portföljvikter för de simulerade finansiella tillgångarna för bolaget. För att modellera framtida skulder för bolaget konsturerades en representativ ITP 2 tjänstepensionsportfölj där pensionskontrakten prissattes med hjälp av traditionell prissättningsteori för livförsäkringar i kontinuerlig tid. Aktuariella antaganden och en konsolideringspolicy konsturerades i linje med de största traditionella livförsäkringsbolagen i Sverige för att konsturea en representativ portfölj. Simuleringarna av bolaget resulterade I kassaöden och utvecklingen av livförsäkringsmatematiska funktioner månadsvis samt solvenskapitalkravet årsvis. Solvenskapitalkravet beräknades med standardformeln erhållen av EIOPA där modulen för marknadsrisk dominerar i bidraget till kapitalkravet. Genom att jämföra det nya riskkänsliga kapitalkravet med solvenskapitalkravet baserat på tidigare regelverk observerades en skillnad i struktur beroende på tid och tillgångsallokering. Storleken på Solvens II-kapitalkravet för livförsäkringsbolag är mer beroende på den finansiella tillgångsstrategin för bolagen medan detta inte är fallet för Solvens I-kapitalkravet. Strukturen på det nya kapitalkravet följer samma struktur som modulen för marknadsrisk där det observerades at lågriskportföljer nödvändigtvis inte motsvarar ett lägre kapitalkrav för livförsäkringsbolaget. Slutsatsen av projektet var att utrymmet för investeringsmöjligheter för svenska livförsäkringsbolag har förminskats. Detta är på grund av införandet av Solvens II då regelverket är mer riskkänsligt än tidigare regelverk.

This paper attempts to study GARCH-type models, with emphasis on fitting GARCH models to exchange rate return series. The symmetric GARCH(1,1) model is compared with the asymmetric EGARCH(1,1) model. Both models are analysed with different conditional distributions, namely Normal, Student's t and skew Student's t for the return innovation. Parameter estimation is performed using a maximum-likelihood approximation. The model performance is assessed by looking at the lowest AIC and BIC. Four exchange rate returns are studied using daily data over the period from 2002 to 2015. Moreover, essential ideas of return time series and stylised facts will be analysed. Our results indicate that the asymmetric GARCH model improves generally estimation with fat-tailed densities in the conditional variance. Furthermore, persistence has found to be reduced with the use of heavy-tailed distributions. Asymmetry presence has been detected in the EGARCH model. Besides, we found that "good news" tend to increase volatility in comparison with "bad news". / Syftet med uppsatsen är att studera modeller av GARCH-typ, och fokus ligger på att anpassa GARCH-modeller efter växelkurstidsserier. Den symmetriska GARCH(1,1)-modellen jämförs med den asymmetriska EGARCH(1,1)-modellen. Modellerna analyseras för olika fördelningar, såsom normal- och t-fördelning, på avkastningarnas brustermer.  För att estimera parametrarna används maximum likelihood-metoden. Modellens prestanda bedöms sedan utifrån AIC- och BIC-kriterierna. Studien är baserad på daglig data från fyra valutapar under perioden 2002 till 2015. Resultaten indikerar att den asymmetriska GARCH-modellen förbättrar estimeringen generellt sett. Genom att använda tjocksvansade fördelningar finner man att persistensen minskar. EGARCH-modellen fångar dessutom upp asymmetrier i avkastningarna, på så sätt att volatiliteten ökar mer vid "goda nyheter" än vid "dåliga nyheter".