[pt] Minerar bitcoins é uma atividade incerta, e para realizá-la, os participantes competem em um processo chamado Proof-Of-Work. Cada participante pode passar meses ou até anos sem fluxos positivos de caixa, enquanto os custos se mantém. Isto pode afastá-los da tecnologia e a saída de membros afeta a própria rede, que não sobrevive sem a presença de mineradores. Este trabalho propõe estudar o compartilhamento de recompensas em estruturas já existentes na rede: mineradores se juntando em pools de mineração e dividindo receitas e custos, assim diminuindo a variabilidade e gerando fluxos positivos de caixa mais constantes. A receita e custos são modelados, e um modelo de programação estocástica é proposto para encontrar as alocações ótimas que garantem a permanência dos membros no pool. Este grupo de é caracterizado por uma coalizão, estudado através de Teoria dos Jogos. O comportamento dos jogadores também é de estudo neste trabalho, e uma medida monetária de risco, na forma de CVaR (Conditional Value at Risk) é usada para representar o perfil de risco do minerador e as consequências para as alocações ótimas. Embora não haja benefício estrito em fazer parte do pool para um único período de análise, há ganho financeiro quando se analisa em múltiplos períodos, e o tempo médio para se acertar um hash diminui quando os participantes se juntam em um pool. Um ganho na probabilidade de mineração ao fazer parte de um pool aumentaria a receita média da coalizão, trazendo ganhos financeiros mesmo em um único período de análise. Divisões intuitivas de recursos, como por poder computacional ou igualitária podem não garantir estabilidade do pool, principalmente considerando períodos longos de tempo. Tal estabilidade é possível em um futuro sem receitas fixas de mineração, se ocorrerem também mudanças nas receitas variáveis e custos. Três funções objetivo diferentes representando três idéias de partilha de recompensa são comparadas e uma metodologia é proposta para uso conjunto de pelo menos duas destas, com objetivo de aumentar a justiça na divisão das recompensas. / [en] Mining Bitcoins is an uncertain activity, and to perform it, players must compete in a process known as Proof-Of-Work. A miner may spend months or even years without positive cash flows on this process, while still incurring in the associated costs. This outcome has the possibility to drive them away from the technology, and the departure of members affects the network itself, as it cannot survive without the presence of miners. This work proposes to study the sharing of rewards in structures already presented in the network: miners joining forces and taking place in mining pools, sharing revenues and costs, thus having positive cash flows more often, reducing variability in gains. The revenues and costs are modeled, and a stochastic optimization model is proposed to find the optimal allocations that guarantee that all members stay within the pool. This group of miners is characterized by a coalition, studied through Game Theory. The behavior of the players is also subject of this study, and a monetary risk measure, by the form of CVaR (Conditional Value at Risk) is used to represent the miner s risk profile and consequences to the optimal allocations. While there is no strict benefit from being part of a pool for a single block, there is financial gain when looking at multi-period, and the average time to correctly guess a hash decreases when players join forces in a pool. A gain in mining probability by being in the pool would raise the average reward of the coalition and allow for financial benefit even in single period.We observe that intuitive sharing allocations such as through computational power and equally dividing rewards may not guarantee the stability of the pool, mainly when longer periods of time are considered. Said stability is possible in the future without fixed incomes, but with changes to the variable rewards and the costs of mining. Lastly, three different objective functions representing three ideas to share the rewards within the nucleolus are compared and a method is proposed to collectively use at least two of them, aiming increased fairness in the sharing of rewards.

In order to ensure solvency, financial institutions must evaluate their credit risk exposure and determine how much economic capital is required to hold as a cushion. This thesis compares three factor models, namely Asymptotic Single Risk Factor (“ASRF”), Inter-sector and Intra-sector factor models and evaluates how their different characteristics affect the economic capital outcomes. The thesis also investigates how these outcomes are affected when assuming asset dependency through a Student's-$t$ copula. Focus will also be put on how different types and levels of asset correlation affect the models' credit risk results. We use a fictitious loan portfolio consisting of 138 Swedish firms with equity data from between 2007 and 2019 in order to calculate asset correlations and economic capital. Our main findings are that the asset correlations severely impact the outcomes of the credit risk models and that practitioners must calibrate and stress test their models regularly with respect to how correlations vary between different firms. The thesis also finds that using copulas for credit portfolios provides more conservative risk outcomes but makes the models less sensitive to correlation level input. / För att finansiella institutioner ska försäkra sig om att vara solventa måste de utvärdera sin exponering mot kreditrisk och därmed avgöra hur mycket ekonomiskt kapital de behöver hålla som buffert. Denna uppsats jämför tre faktormodeller vid namn Asymptotic Systematic Risk Factor (“ASRF”), Inter-sektor, och Intra-sektor med syfte att undersöka hur deras olika karaktärsdrag påverkar estimaten för ekonomiskt kapital. Vi utvärderar även hur utfallen påverkas av införandet av copula-beroende mellan portföljtillgångarna. Fokus kommer även att läggas på hur olika typer och nivåer av korrelation mellan bolag påverkar de olika modellernas kreditriskutfall. Vi använder oss av en fiktiv låneportfölj bestående av 138 svenska bolag med aktieprisdata mellan 2007 och 2019 för att beräkna korrelationer och ekonomiskt kapital. Uppsatsens främsta resultat pekar på att korrelationerna har en väldigt stor påverkan på det ekonomiska kapitalet och att analytiker rekommenderas att kontinuerligt kalibrera och stresstesta sina modeller med avseende på hur korrelationerna kan skilja sig mellan olika bolag. Vi fann även att copula-beroende gav mycket mer konservativa utfall, det vill säga ett högre ekonomiskt kapital, men var mindre känslig för korrelationsnivåerna mellan bolagen i portföljen.

The demand for sustainable investments has seen an increase in recent years. There is considerable literature covering backtesting of the performance and risk of socially responsible investments (SRI) compared to conventional investments. However, literature that models and examines the risk characteristics of SRI compared to conventional investments is limited. This thesis seeks to model and compare the risk of mutual funds scoring in the top 10% in terms of sustainability, based on Morningstar Portfolio Sustainability Score, to those scoring in the bottom 10%. We create one portfolio consisting of the top 10% funds and one portfolio consisting of the bottom 10%, for European and global mutual funds separately, thus in total creating 4 portfolios. The analysis is based on data of the funds' returns and Morningstar Portfolio Sustainability Scores during December 2015 to August 2019. Investigating several GARCH models, we find an ARMA-GARCH model with skewed Student's t-distribution as innovation distribution to give the best fit to the daily log-returns of each portfolio. Based on the fitted ARMA-GARCH models with skewed Student's t-distribution, we use a parametric bootstrap method to compute 95% confidence intervals for the difference in long-run volatility and value at risk (VaR) between the portfolios with high and low Morningstar Portfolio Sustainability Scores. This is performed on the portfolios of European and global funds separately. We conclude that, for global and European funds respectively, no significant difference in terms of long-run volatility and VaR is found between the funds in each of the 10% ends of the Morningstar Portfolio Sustainability Score. / Efterfrågan av hållbara investeringar har ökat kraftigt de senaste åren. Det finns många studier som genomför backtesting av hållbara investeringars avkastning och risk jämfört med konventionella investeringar. Färre studier har däremot gjorts för att modellera och jämföra investeringarnas riskegenskaper. Denna uppsats syftar till att modellera risken av hållbara investeringar genom att jämföra de 10% fonder med högst Morningstar Portfolio Sustainability Score mot de 10% fonder med lägst score. Jämförelsen görs separat för globala fonder och europeiska fonder, vilket resulterar i totalt 4 portföljer. Analysen baseras på data på fondernas avkasting och Morningstar Portfolio Sustainability Score under tidsperioden december 2015 till augusti 2019. Genom att undersöka flera olika GARCH-modeller, kommer vi fram till att en ARMA-GARCH-modell med skev t-fördelning bäst beskriver den dagliga logaritmerade avkastningen för varje portfölj. Baserat på de anpassade ARMA-GARCH-modellerna, används en "parametric bootstrap"-metod för att beräkna 95%-iga konfidensintervall för skillnaden i långsiktig volatilitet och value at risk (VaR) mellan portföljerna med högt och lågt Morningstar Portfolio Sustainability Score. Detta görs separat för de europeiska och globala fonderna. Vår slutsats är att det, för globala och europeiska fonder, inte råder en signifikant skillnad i långsiktig volatilitet eller VaR mellan fonder med högt och lågt Morningstar Portfolio Sustainability Score.

Allocation to private infrastructure debt investments has increased in the recent years [15]. For managers of multi-asset portfolios, it is important to be able to assess the risk of the total portfolio and the contribution to risk of the various holdings in the portfolio. This includes being able to explain the risk of having private infrastructure debt investments in the portfolio. The modelling of private infrastructure debt face many challenges, such as the lack of private data and public indices for private infrastructure debt. In this thesis, two approaches for modelling private infrastructure debt in a parametric risk factor model are proposed. Both approaches aim to incorporate revenue risk, which is the risk occurring from the type of revenue model in the infrastructure project or company. Revenue risk is categorised into three revenue models; merchant, contracted and regulated, as spread level differences can be distinguished for private infrastructure debt investments using this categorisation. The difference in spread levels between the categories are used to estimate β coefficients for the two modelling approaches. The spread levels are obtained from a data set and from a previous study. In the first modelling approach, the systematic risk factor approach, three systematic risk factors are introduced where each factor represent infrastructure debt investments with a certain revenue model. The risk or the volatility for each of these factors is the volatility of a general infrastructure debt index adjusted with one of the β coefficients. In the second modelling approach, the idiosyncratic risk term approach, three constant risk terms for the revenue models are added in order to capture the revenue risk for private infrastructure debt investments. These constant risk terms are estimated with the β coefficients and the historical volatility of a infrastructure debt index. For each modelling approach, the commonly used risk measures standalone risk and risk contribution are presented for the entire block of the infrastructure debt specific factors and for each of the individual factors within this block. Both modelling approaches should enable for better explanation of risk in private infrastructure debt investments by introducing revenue risk. However, the modelling approaches have not been backtested and therefore no conclusion can be made in regards to whether one of the proposed modelling approaches actually is better than current modelling approaches for private infrastructure debt. / Investeringar i privat infrastrukturskuld har ökat de senaste åren [15]. För βägare av portföljer med investeringar i samtliga tillgångsslag är det viktigt att kunna urskilja risken från de olika innehaven i portföljen. Det finns många utmaningar vad gäller modellering av privat infrastrukturskuld, så som den begränsade mängden privat data och publika index för privat infrastrukturskuld. I denna uppsats föreslås två tillvägagångssätt för att modellera privat infrastrukturskuld i en parametrisk riskfaktormodell. Båda tillvägagångssätten eftersträvar att inkorporera intäktsrisk, vilket är risken som beror på den underliggande intäktsmodellen i ett infrastrukturprojekt eller företag. Intäksrisk delas in i intäksmodellerna "merchant", "contracted" och "regulated", då en skillnad i spreadnivå mellan privata infrastrukturskuldinvesteringar kan urskiljas med denna kategorisering. Skillnaden i spreadnivå mellan de olika kategorierna används för att estimera β -koefficienter som används i båda tillvägagångssätten. Spreadnivåerna erhålls från ett dataset och från en tidigare studie. I det första tillvägagångssättet, den systematiska riskfaktor-ansatsen, introduceras tre systematiska riskfaktorer som representerar infrastrukturskuldinvesteringar med en viss intäktsmodell. Risken eller volatiliten för dessa faktorer är densamma som volatiliteten för ett index för infrastrukturskuld justerat med en av β -koefficienterna. I det andra tillvägagångssättet, den idriosynktratiska riskterm-ansatsen, adderas tre konstanta risktermer för intäktsmodellerna för att fånga upp intäktsrisken i de privata infrastrukturinvesteringarna. De konstanta risktermerna är estimerade med β -koefficienterna och en historisk volatilitet för ett index för infrastrukturskuld. För båda tillvägagångssätten presenteras riskmåtten stand-alone risk1 och risk contribution2. Riskmåtten ges för ett block av samtliga faktorer för infrastrukturskuld och för varje enskild faktor inom detta block. Båda tillvägagångssätten borde möjliggöra bättre förklaring av risken för privata infrastrukturskuldinvesteringar i en större portfölj genom att ta hänsyn till intäktsrisken. De två tillvägagångssätten för modelleringen har dock ej testats. Därför kan ingen slutsats dras med hänsyn till huruvida ett av tillvägagångssätten är bättre än de som används för närvärande för modellering av privat infrastrukturskuld.

[pt] À medida que o mundo enfrenta a urgente questão das mudanças climáticas, a energia eólica se destaca como uma fonte crítica de energia limpa. No entanto, realizar seu pleno potencial depende da otimização dos layouts de parques eólicos, especialmente à luz do complexo efeito de esteira. Esta dissertação adentra na Otimização de Layout de Parques Eólicos (WFLO, na sigla em inglês) usando o Modelo de Efeito de Esteira de Bastankhah. O escopo deste estudo vai além do design de layout; abrange a intrincada tarefa de mitigar o impacto do efeito de esteira, juntamente com a busca por uma estratégia de negociação com aversão ao risco e maximização de valor. Para contabilizar a aversão ao risco, uma combinação entre o Valor Esperado e os funcionais de medida de risco baseados no quantil esquerdo, a medida de Valor em Risco Condicional (CVaR). Para apoiar esta pesquisa, um pacote de código aberto OptimalLayout.jl foi desenvolvido. Este pacote co-otimiza o posicionamento das turbinas eólicas para mitigar o impacto do efeito de esteira e a estratégia de contratação de um agente/gerador avesso ao risco. Através de uma série de estudos de casos práticos em diversos ambientes dinâmicos, esta pesquisa ilustra a aplicabilidade do WFLO no mundo real. Estas investigações examinam detalhadamente a sua influência na produção de energia e na dinâmica das receitas, oferecendo informações valiosas sobre soluções energéticas sustentáveis. / [en] As the world confronts the pressing issue of climate change, wind power stands out as a critical source of clean energy. However, realizing its full potential relies on the optimization of wind farm layouts, particularly in light of the complex wake effect. This dissertation delves into Wind Farm Layout Optimization (WFLO) using the Bastankhah Wake Model. The scope of this study goes beyond layout design; it encompasses the intricate task of mitigating the wake effect s impact along with the seek for a risk-averse-value maximizing trading strategy. To account for risk-averseness, a combination between Expected Value and the left-side-quantile-based risk-measure functionals, the Conditional Value-at-Risk (CVaR) measure. To support this research, an opensource package OptimalLayout.jl was developed. This package co-optimizes the positioning of wind turbines to mitigate wake effect impact,and the contracting strategy of a Risk-Averse agent/generator. Through a series of practical case studies across diverse dynamic environments, this research illustrates the real-world applicability of WFLO. These investigations intricately examine its influence on power production and revenue dynamics, offering valuable insights into sustainable energy solutions.

This thesis examines the risk profile of Electronically Traded Funds and the dependence of the ESG rating on risk. 527 ETFs with exposure globally were analyzed. Risk measures considered were Value-at-Risk and Expected Shortfall, while some other metrics of risk was used, such as the volatility, maximum drawdown, tail dependece, and copulas. Stress tests were conducted in order to test the resilience against market downturns. The ETFs were grouped by their ESG rating as well as by their carbon intensity. The results show that the lowest risk can be found for ETFs with either the lowest ESG rating or the highest. Generally, a higher ESG rating implies a lower risk, but without statistical significance in many cases. Further, ETFs with a higher ESG rating showed, on average, a lower maximum drawdown, a higher tail dependence, and more resilience in market downturns. Regarding volatility, the average was shown to be lower on average for ETFs with a higher ESG rating, but no statistical significance could be found. Interestingly, the results show that investing sustainably returns a better financial performance at a lower risk, thus going against the Capital Asset Pricing Model. / Denna studie undersöker riskprofilen för elektroniskt handlade fonder och sambandet mellan risk och hållbarhetsbetyg. 527 ETF:er med global exponering analyserades. De riskmått som användes var Value-at-Risk och Expected Shortfall, och några andra mått för risk användes, däribland volatilitet, största intradagsnedgång, samband i svansfördelning, och copulas. Stresstest utfördes för att testa motsåtndskraften i marknadsnedgångar. ETF:erna grupperades med hjälp av deras hållbarhetsbetyg och deras koldioxidintensitet. Resultatet visar att lägst risk finns i ETF:er med högst respektive lägst hållbarhetsbetyg. Generellt har ETF:er med högre hållbarhetsbetyg en lägre risk, med endast viss statistisk signifikans. Därtill har ETF:er med högre hållbarhetsbetyg, i genomsnitt, en lägre största intradagsnedgång, högre samband i fördelningssvansarna och är mer motståndskraftiga i marknadsnedgångar. Volatiliteten är i genomsnitt lägre desto högre hållbarhetsbetyget är, men detta resultat saknar statistisk signifikans. Ett intressant resultat är att om man investerar hållbart kan man få en högre avkastning med en lägre risk, vilket går emot Capital Asset Pricing Model.

Asset management within the nuclear industry has become an increasingly relevant topic as safety requirements have tightened and energy security has become more important. Asset management ensures performance and reliability in a nuclear facility by balancing costs, opportunities, and risks to get the most out of assets. Asset management processes can often be modeled as capital budgeting problems, where investments are evaluated based on costs and benefits, which establishes a link to mathematical optimization. This study addresses asset management at the Swedish nuclear power plant, Forsmark, and aims to implement an optimization model to improve the project selection related to maintenance and replacement of assets at the plant. First, the most relevant areas of nuclear asset management are identified through a comprehensive literature review. The most relevant method, identified as a mix between risk-informed asset management and capital budgeting, is then implemented to fit the prerequisites at Forsmark. Several models of different complexity are developed and the resulting stochastic model incorporates uncertainty of input variables by assuming underlying distributions. Finally, a methodology to incorporate a quantitative risk measure in the optimization is suggested through the use of conditional value at risk. Results are generated based on simulated data and illustrate the potential of implementing the method at Forsmark. / Tillgångsförvaltning inom kärnkraftsindustrin har blivit alltmer aktuellt i takt med att säkerhetskraven har skärpts och tillförlitlighet i energiproduktionen blivit viktigare. Effektiv tillgångshantering säkerställer prestanda och reliabilitet i ett kärnkraftverk genom att hitta en balans mellan kostnader, möjligheter och risker för att maximera nyttan av tillgångar. Projekturval i tillgångsförvaltningen kan ofta modelleras som ett kapitalbudgeteringsproblem, där investeringar utvärderas utifrån kostnader och uppsida, vilket påvisar en koppling till matematisk optimering. Denna studie behandlar tillgångshantering vid det svenska kärnkraftverket Forsmark och syftar till att implementera en optimeringsmodell för att förbättra projekturvalet relaterat till underhåll av tillgångar vid anläggningen. Det första steget i studien bearbetar den befintliga litteraturen inom området för att få en uppfattning av relevanta metoder. Den mest relevanta metoden identifierades som en mix mellan riskinformerad tillgångsförvaltning och kapitalbudgetering. En metod baserad på de generella principerna för dessa områden utvecklades och anpassades för de specifika förutsättningarna på Forsmark. Flera modeller av olika komplexitet utvecklades och den slutgiltiga stokastiska modellen inkorporerar osäkerhet i de mest relevanta ingångsvariablerna genom att anta sannolikhetsfördelningar. Slutligen föreslås en metod för att implementera ett kvantitativt riskmått i optimeringen genom att använda CVaR. Resultaten genereras utifrån simulerade data och illustrerar potentialen i att implementera metoden på Forsmark.

This thesis investigates applying the semiparametric method Peaks-Over-Threshold on data generated from a Monte Carlo simulation when estimating the financial risk measures Value-at-Risk and Expected Shortfall. The goal is to achieve a faster convergence than a Monte Carlo simulation when assessing extreme events that symbolise the worst outcomes of a financial portfolio. Achieving a faster convergence will enable a reduction of iterations in the Monte Carlo simulation, thus enabling a more efficient way of estimating risk measures for the portfolio manager.  The financial portfolio consists of US life insurance policies offered on the secondary market, gathered by our partner RessCapital. The method is evaluated on three different portfolios with different defining characteristics.  In Part I an analysis of selecting an optimal threshold is made. The accuracy and precision of Peaks-Over-Threshold is compared to the Monte Carlo simulation with 10,000 iterations, using a simulation of 100,000 iterations as the reference value. Depending on the risk measure and the percentile of interest, different optimal thresholds are selected.  Part II presents the result with the optimal thresholds from Part I. One can conclude that Peaks-Over-Threshold performed significantly better than a Monte Carlo simulation for Value-at-Risk with 10,000 iterations. The results for Expected Shortfall did not achieve a clear improvement in terms of precision, but it did show improvement in terms of accuracy.  Value-at-Risk and Expected Shortfall at the 99.5th percentile achieved a greater error reduction than at the 99th. The result therefore aligned well with theory, as the more "rare" event considered, the better the Peaks-Over-Threshold method performed.  In conclusion, the method of applying Peaks-Over-Threshold can be proven useful when looking to reduce the number of iterations since it do increase the convergence of a Monte Carlo simulation. The result is however dependent on the rarity of the event of interest, and the level of precision/accuracy required. / Det här examensarbetet tillämpar metoden Peaks-Over-Threshold på data genererat från en Monte Carlo simulering för att estimera de finansiella riskmåtten Value-at-Risk och Expected Shortfall. Målet med arbetet är att uppnå en snabbare konvergens jämfört med en Monte Carlo simulering när intresset är s.k. extrema händelser som symboliserar de värsta utfallen för en finansiell portfölj. Uppnås en snabbare konvergens kan antalet iterationer i simuleringen minskas, vilket möjliggör ett mer effektivt sätt att estimera riskmåtten för portföljförvaltaren.  Den finansiella portföljen består av amerikanska livförsäkringskontrakt som har erbjudits på andrahandsmarknaden, insamlat av vår partner RessCapital. Metoden utvärderas på tre olika portföljer med olika karaktär.  I Del I så utförs en analys för att välja en optimal tröskel för Peaks-Over-Threshold. Noggrannheten och precisionen för Peaks-Over-Threshold jämförs med en Monte Carlo simulering med 10,000 iterationer, där en Monte Carlo simulering med 100,000 iterationer används som referensvärde. Beroende på riskmått samt vilken percentil som är av intresse så väljs olika trösklar.  I Del II presenteras resultaten med de "optimalt" valda trösklarna från Del I. Peaks-over-Threshold påvisade signifikant bättre resultat för Value-at-Risk jämfört med Monte Carlo simuleringen med 10,000 iterationer. Resultaten för Expected Shortfall påvisade inte en signifikant förbättring sett till precision, men visade förbättring sett till noggrannhet.  För både Value-at-Risk och Expected Shortfall uppnådde Peaks-Over-Threshold en större felminskning vid 99.5:e percentilen jämfört med den 99:e. Resultaten var därför i linje med de teoretiska förväntningarna då en högre percentil motsvarar ett extremare event.  Sammanfattningsvis så kan metoden Peaks-Over-Threshold vara användbar när det kommer till att minska antalet iterationer i en Monte Carlo simulering då resultatet visade att Peaks-Over-Threshold appliceringen accelererar Monte Carlon simuleringens konvergens. Resultatet är dock starkt beroende av det undersökta eventets sannolikhet, samt precision- och noggrannhetskravet.

In den letzten Jahren gab es ein drastisches Wachstum in verfügbaren Finanzdaten. Finanzmärkte haben starke und oft nicht ganz vorhersagbare Änderungen ihrer Dynamik erlebt. Diese Tendenz hat dazu geführt, dass die Methoden der Risikomodellierung sowohl das Problem der hohen Dimensionalität als auch dynamische nicht Gaußsche Strukturen behandeln müssen. Das Ziel dieser Dissertation ist es, Methoden der Risikomodellierung vorzuschlagen, die gleichzeitig Reduzierung der Dimensionalität und dynamische Struktur in drei Anwendungen erlauben: 1) Asset Allocation und Hedging, 2) stochastische Modellierung von multivariaten Prozessen, 2) Messung der systemischen Risiken. Die vorgeschlagenen Methoden demonstrieren gute Ergebnisse im Vergleich mit den existierenden Methoden der Risikomodellierung und führen neue Verfahren zur Erkennung der extremen Risiken und Anomalien auf Finanzmärkten sowie zur deren Management. / Over the recent years, there have been a significant increase in financial data availability. On the other hand, financial markets have experienced sharp and often unforeseen changes in their dynamics. This tendency has caused the need for risk modeling approaches addressing both high dimensionality problem and accustoming for dynamic non Gaussian structure. The primary aim of this dissertation is to propose several risk modeling approaches which allow for simultaneous dimension reduction and dynamic structures in three setups: 1) asset allocation and hedging, 2) stochastic surface modeling and 3) systemic risk determination. Proposed models demonstrate good performance when compared to existing approaches for risk modeling and introduce new flexible ways to detect extreme risks and anomalies on financial markets as well as methods for their modeling and management.

<p>Optimization under uncertainty is the field of optimization where the data or the optimization model itself has uncertainties associated with it. Such problems are more commonly referred to as stochastic optimization problems. These problems capture the broad idea of making optimal decisions under uncertainty. An important class of these stochastic optimization problems is chance-constrained optimization problems, where the decision maker seeks to choose the best decision such that the probability of violating a set of uncertainty constraints is within a predefined probabilistic threshold risk level. Such stochastic optimization problems have found a lot of interest in the service industry as the service providers need to satisfy a minimum service level agreement (SLA) with their customers. Satisfying SLA in the presence of uncertainty in the form of probabilistic failure of infrastructure poses many interesting and challenging questions. In this thesis, we answer a few of these questions.</p> <p>We first explore the problem of quantifying uncertainties that adversely impact the service provider's infrastructure, thereby hurting the service level agreements. In particular we address the probability quantification problem, where given an uncertainty set, the goal is to quantify the probability of an event, on which the optimal value of an optimization problem exceeds a predefined threshold value. The novel techniques we propose, use and develop ideas from diverse literatures such as mixed integer nonlinear program, chance-constrained programming, approximate sampling and counting techniques, and large deviation bounds. Our approach yields the first polynomial time approximation scheme for the specific probability quantification problem we consider. </p> <p>Our next work is inspired by the ideas of risk averse decision making. Here, we focus on studying the problem of minimizing risk functions. As a special case we also explore the problem of minimizing the Value at Risk (VaR), which is a well know non-convex problem. For more than a decade, the well-known, best convex approximation to this problem has been obtained by minimizing the Conditional Value at Risk (CVaR). We proposed a new two-stage model which formulates these risk functions, which eventually leads to a bilinear optimization problem, a special case of which is the VaR minimization problem. We come up with enhancements to this bilinear formulation and use convexification techniques to obtain tighter lower and upper convex approximations to the problem. We also find that the approximation obtained by CVaR minimization is a special case of our method. The overestimates we construct help us to develop tighter convex inner approximations for the chance constraint optimization problems.</p>