This thesis focuses on stochastic dominance in portfolio selection problems. The thesis recalls basic knowledge from the area of portfolio optimization with utility functions and first, second, $N$-th and infinite order of stochastic dominance. It sumarizes Post's, Kuosmanen's and Kopa's criteria for portfolio efficiency and necessary and sufficient conditions of stochastic dominance for discrete and continuous probability distributions. The thesis also contains formulations of optimization problems with second order stochastic dominance constraints derived for discrete and continuous probability distributions. A practical application is also a part of the thesis, where the optimization problems for monthly returns of Czech stocks are solved using optimization software GAMS.

Cette thèse se propose d’analyser l’évolution de la pauvreté et d’identifier les groupes socio-économiques ainsi que les dimensions qui contribuent le plus à l’évolution de la pauvreté en Tunisie en nous appuyant sur une approche monétaire et une approche multidimensionnelle. Pour cela, nous adoptons une méthodologie qui consiste à utiliser l’approche de dominance stochastique et la théorie des ensembles flous, c’est-à-dire des méthodes différentes de celles adoptées dans les études antérieures qui s’intéressent à la Tunisie. Trois principales conclusions ressortent de l’exploitation des données issues de deux enquêtes nationales sur le budget et la consommation des ménages 2005, 2010 et d’une enquête nationale sur la santé de la famille 2006. La première révèle une diminution de la pauvreté au niveau national bien que de fortes disparités persistent encore entre le milieu rural et le milieu urbain, et entre les régions du littoral et les régions de l’intérieur. La deuxième met en évidence les effets des politiques de redistribution en Tunisie qui n’ont pas permis d’accélérer le rythme de réduction de la pauvreté. La troisième suggère que la prise en compte de l’aspect multidimensionnel de la pauvreté révèle l’existence d’autres dimensions importantes en relation avec la pauvreté et qui constituent un obstacle à une vie décente pour les ménages tunisiens. / The aim of this thesis is to analyze the evolution of poverty and identify the socio-economical groups as well as the dimensions that contribute to it in Tunisia following a monetary approach and also a multi-dimensional one. For this reason, I adopt a method which consists in using the approach of stochastic dominance and the theory of fuzzy sets. That is to say, I adopt some of the different methods which have been adopted in the previous studies that were interested in Tunisia. Three main conclusions emerge from the exploitation of data based on two national surveys of the budget and the consumption of households 2005, 2010 and another national survey of the family health 2006. In fact, the first one reveals the decrease of poverty at the national level despite that there are disparities that still persist between rural and urban areas and between the coastal regions and the regions of the interior as well. The second survey puts in evidence that the effects of redistributive policies in Tunisia have not accelerated the pace of poverty reduction. As for the third survey, it suggests that taking into account the multi-dimensional aspect of poverty reveals the existence of other important dimensions in relationship with poverty that constitute an obstacle to a decent life for the Tunisian households.

We use modern approach of stochastic dominance in portfolio optimization, where we want the portfolio to dominate a benchmark. Since the distribution of returns is often just estimated from data, we look for the worst distribution that differs from empirical distribution at maximum by a predefined value. First, we define in what sense the distribution is the worst for the first and second order stochastic dominance. For the second order stochastic dominance, we use two different formulations for the worst case. We derive the robust stochastic dominance test for all the mentioned approaches and find the worst case distribution as the optimal solution of a non-linear maximization problem. Then we derive programs to maximize an objective function over the weights of the portfolio with robust stochastic dominance in constraints. We consider robustness either in returns or in probabilities for both the first and the second order stochastic dominance. To the best of our knowledge nobody was able to derive such program before. We apply all the derived optimization programs to real life data, specifically to returns of assets captured by Dow Jones Industrial Average, and we analyze the problems in detail using optimal solutions of the optimization programs with multiple setups. The portfolios calculated using...

This dissertation contains three applications of time series in finance and macroeconomics. The first essay compares the cumulative returns for stocks and bonds at investment horizons from one to ten years by using a test for spatial dominance. Spatial dominance is a variation of stochastic dominance for nonstationary variables. The results suggest that for investment horizons of one year, bonds spatially dominate stocks. In contrast, for investment horizons longer than five years, stocks spatially dominate bonds. This result is consistent with the advice given by practitioners to long term investors of allocating a higher proportion of stocks in their portfolio decisions. The second essay presents a method that allows testing of whether or not an asset stochastically dominates the other when the time horizon is uncertain. In this setup, the expected utility depends on the distribution of the value of the asset as well as the distribution of the time horizon, which together form the weighted spatial distribution. The testing procedure is based on the Kolmogorov Smirnov distance between the empirical weighted spatial distributions. An empirical application is presented assuming that the event of exit time follows an independent Poisson process with constant intensity. The last essay applies a dynamic factor model to generate out-of-sample forecasts for the inflation rate in Mexico. Factor models are useful to summarize the information contained in large datasets. We evaluate the role of using a wide range of macroeconomic variables to forecast inflation, with particular interest on the importance of using the consumer price index disaggregated data. The data set contains 54 macroeconomic series and 243 consumer price subcomponents from 1988 to 2008. The results indicate that factor models outperform the benchmark autoregressive model at horizons of one, two, four and six quarters. It is also found that using disaggregated price data improves forecasting performance.

Tradeoff studies are a common part of engineering practice. Designers conduct tradeoff studies in order to improve their understanding of how various design considerations relate to one another. Generally a tradeoff study involves a systematic multi-criteria evaluation of various alternatives for a particular system or subsystem. After evaluating these alternatives, designers eliminate those that perform poorly under the given criteria and explore more carefully those that remain. The capability to compose preexisting tradeoff studies is advantageous to the designers of engineered systems, such as aircraft, military equipment, and automobiles. Such systems are comprised of many subsystems for which prior tradeoff studies may exist. System designers conceivably could explore system-level tradeoffs more quickly by leveraging this knowledge. For example, automotive systems engineers could combine tradeoff studies from the engine and transmission subsystems quickly to produce a comprehensive tradeoff study for the power train. This level of knowledge reuse is in keeping with good systems engineering practice. However, existing procedures for generating tradeoff studies under uncertainty involve assumptions that preclude engineers from composing them in a mathematically rigorous way. In uncertain problems, designers can eliminate inferior alternatives using stochastic dominance, which compares the probability distributions defined in the design criteria space. Although this is well-founded mathematically, the procedure can be computationally expensive because it typically entails a sampling-based uncertainty propagation method for each alternative being considered. This thesis describes two novel extensions that permit engineers to compose preexisting subsystem-level tradeoff studies under uncertainty into mathematically valid system-level tradeoff studies and efficiently eliminate inferior alternatives through intelligent sampling. The approaches are based on three key ideas: the use of stochastic dominance methods to enable the tradeoff evaluation when the design criteria are uncertain, the use of parameterized efficient sets to enable reuse and composition of subsystem-level tradeoff studies, and the use of statistical tests in dominance testing to reduce the number of behavioral model evaluations. The approaches are demonstrated in the context of a tradeoff study for a motor vehicle.

The relationship between quantity traded and transaction costs has been one of the main focuses among financial scholars and practitioners. The purpose of this thesis is to investigate the informational relationship between these variables. Following insights and results of Milgrom (1981), Feldman (2004), and Feldman and Winer (2004), we use New York Stock Exchange (NYSE) data and kernel estimation methods to construct the distribution of one variable conditional on the other. We then study the information in these conditional distributions: the extent to which they are ordered by first order stochastic dominance (FOSD) and by monotone likelihood ratio property (MLRP). We find that transaction size and effective spread are statistically significantly orrelated. FOSD, a necessary condition for a "separating signaling equilibrium", holds under certain conditions. We start from two-subsample case. We choose a cut-off point in transaction size and categorize the observations with transaction sizes smaller than the cut-off point into group "low". The remaining data is classified as "high". We repeat this procedure for all possible transaction size cut-off points. It turns out that FOSD holds nowhere. However, once we eliminate transactions at the quote midpoint, the "crossings" between exchange members not specialists, FOSD holds for all the cut-off points fewer than 15800 shares. MLRP, a necessary and sufficient condition for the separating equilibrium to hold point by point of the conditional density functions, does not hold but might not be ruled out considering the error in the estimates. We also find that large trades are not necessarily associated with large spread. Instead, it is more likely that larger trades are transacted at the quote midpoint (again, the non-specialist "crossings") than smaller trades. Our results confirm the findings of Barclay and Warner (1993) regarding the informativeness of medium-size transactions: we identify informational relationships between mid-size transactions and spreads but not for trades at the quote midpoint and large-size transactions. That is, we identify two regimes, an informational one and a non-informational/liquidity one.

Title: Bilevel optimization problems and their applications to portfolio selection Author: Lenka Godul'ová Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D. Abstract: This work deals with the problem of bilevel tasks. First, it recalls the basic knowledge of mean-risk models, risk measure in singlelevel problems, and second degree stochastic dominance. Then it presents basic knowledge of bilevel tasks. bilevel problems have several advantages over singlelevel. In one process, it is possible to analyze two different or even conflicting situations. The bilevel role can better capture the relationship between the two objects. The main focus of the thesis is the formulation of various bilevel tasks and their reformulation into the simplest form. The numerical part deals with four types of formulated bilevel problems at selected risk measures. Keywords: Bilevel problems, Second degree stochastic dominance, Risk measures 1

Title: Bilevel optimization problems and their applications to portfolio selection Author: Lenka Godul'ová Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D. Abstract: This work deals with the problem of bilevel tasks. First, it recalls the basic knowledge of mean-risk models, risk measure in singlelevel problems, and second degree stochastic dominance. Then it presents basic knowledge of bilevel tasks. bilevel problems have several advantages over singlelevel. In one process, it is possible to analyze two different or even conflicting situations. The bilevel role can better capture the relationship between the two objects. The main focus of the thesis is the formulation of various bilevel tasks and their reformulation into the simplest form. The numerical part deals with four types of formulated bilevel problems at selected risk measures. Keywords: Bilevel problems, Second degree stochastic dominance, Risk measures 1

Pøedlo¾ená práce se zabývá dvìma pøístupy øe¹ení problému volby portfolia. Prvním jsou čmean-riskÿ modely, které minimalizují riziko pro pøedem zvolený výnos nebo maximalizují výnos pro pevnì stanovené riziko. Druhým je princip stochastické dominance, úzce související s teorií u¾itku. Cílem této diplomové práce je zkoumat vztah mezi mno¾inami e cientních portfolií, které jsou øe¹e- ním v obou pøístupech. Pro kvanti kaci rizika se kromì základních mìr jako jsou rozptyl, V aR nebo CV aR v práci uva¾ují i spektrální míry, zohledòující sub- jektivní postoj investora k riziku. Uká¾eme, za platnosti jakých podmínek jsou modely minimalizující spektrální míry konzistentní se stochastickou dominancí druhého øádu (SSD). Aplikujeme Kopa-Postùv test, který je jedním z více testù na SSD e cienci portfolia, na reálná data z americké burzy cenných papírù a SSD e cientní portfolia porovnáme s e cientními portfoliami získanými minimalizací CV aR-u uva¾ovaného na rùznych hladinách spolehlivosti. 1

In this work we introduce some category-theoretical concepts and techniques to study probability distributions on metric spaces and ordered metric spaces. In Chapter 1 we give an overview of the concept of a probability monad, first defined by Giry. Probability monads can be interpreted as a categorical tool to talk about random elements of a space X. We can consider these random elements as formal convex combinations, or mixtures, of elements of X. Spaces where the convex combinations can be actually evaluated are called algebras of the probability monad. In Chapter 2 we define a probability monad on the category of complete metric spaces and 1-Lipschitz maps called the Kantorovich monad, extending a previous construction due to van Breugel. This monad assigns to each complete metric space X its Wasserstein space PX. It is well-known that finitely supported probability measures with rational coefficients, or empirical distributions of finite sequences, are dense in the Wasserstein space. This density property can be translated into categorical language as a colimit of a diagram involving certain powers of X. The monad structure of P, and in particular the integration map, is uniquely determined by this universal property. We prove that the algebras of the Kantorovich monad are exactly the closed convex subsets of Banach spaces. In Chapter 3 we extend the Kantorovich monad of Chapter 2 to metric spaces equipped with a partial order. The order is inherited by the Wasserstein space, and is called the stochastic order. Differently from most approaches in the literature, we define a compatibility condition of the order with the metric itself, rather then with the topology it induces. We call the spaces with this property L-ordered spaces. On L-ordered spaces, the stochastic order induced on the Wasserstein spaces satisfies itself a form of Kantorovich duality. The Kantorovich monad can be extended to the category of L-ordered metric spaces. We prove that its algebras are the closed convex subsets of ordered Banach spaces, i.e. Banach spaces equipped with a closed cone. The category of L-ordered metric spaces can be considered a 2-category, in which we can describe concave and convex maps categorically as the lax and oplax morphisms of algebras. In Chapter 4 we develop a new categorical formalism to describe operations evaluated partially. We prove that partial evaluations for the Kantorovich monad, or partial expectations, define a closed partial order on the Wasserstein space PA over every algebra A, and that the resulting ordered space is itself an algebra. We prove that, for the Kantorovich monad, these partial expectations correspond to conditional expectations in distribution. Finally, we study the relation between these partial evaluation orders and convex functions. We prove a general duality theorem extending the well-known duality between convex functions and conditional expectations to general ordered Banach spaces.