Convex set

Luke Stein Stanford graduate economics core (2006–7) Reference and review (last updated September 29, 2009) Contents 1 Econometrics . . . . . . . . . . . . . . 1.1 Probability foundations . . . . . . . . Basic set theory Disjointness Partition Sigma algebra Probability function Probability space Counting Conditional probability Bayes’ Rule Independence of events Mutual independence of events Random variables . . . . . . . . . . . Random variable Random vector Measurability Smallest σ-field Independence

Arthur CHARPENTIER - econometrie de la finance (2008/2009) The Basel II accord (2004) June 2004, the Basel Committee finalized the Basel Accords, based on three pillars • minimum regulatory requierements, i.e. some risk-based capital requirements : set capital charges against credit risk (internal rating based), market risk (internal model approach) and operational risk. the goal is to keep constant the level of capital in the global banking syste : 8% of risk weighted assets, • supervisorv review

. . . . . . . . . . . . . . . . . . . . . . . . . . . The framework 3.2.1 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 56 57 58 59 60 62 64 66 66 67 68 69 69 73 The set of nancial positions Absence of arbitrage in the nancial market . . . . . . . . . . . . . Admissible portfolios and closedness property . . . . . . . . . . . . Illustration of the framework by examples of nancial markets . . . Addition of production

morceau de phrase a le tle principal et un autre morceau de phrase a le rdle secondaire, C'est la base ajoute pour compléter a base Proposition principale] + CusorSoma) + |Proposition subordonnée| Tie comport un vrbe conve le compart un verse cong ou Ti Ce qui distingue les subordonnées, c'est ce qu'elles complétent : Je si elle complete un nom : c'est une proposition subordonnée relative «si elle compléte toute une phrase : c'est une proposition subordonnée

yield of equal change. We can see it well on the graph with the bond D, for instance while the yield decrease of 5%, bond’s price increase around 150% but while the yield increase to 5%, bond’s price decrease only around 50%. Because here the curve is convex, which mean decreases in yields have bigger impact on price than increases in yield of same range. Price of long-term bonds is more sensitive to interest rate than price of short-term bond. It’s normal because the impact of higher discount rate will

strategies [3], media selection [16 and 20], setting acquisition programs [3 and 13], and setting optimal promotion budgets [6]. The purpose of this paper is to offer another managerial application of customer equity, that of optimally allocating an already set promotion budget under different market conditions, focusing on the acquisition/retention allocation. A major contribution of this paper is: (1) its development of a general approach to the optimal allocation of promotion budget through a combination

theorem states: Let S be a non-empty, compact and convex subset of some Euclidean space Rn. Let φ: S → 2S be a set-valued function on S with a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then φ has a fixed point. Closed graph A set-valued function φ: X→2Y is said to have a closed graph if the set {(x,y)| y ∈ φ(x)} is a closed subset of X×Y in the product topology. Fixed point Let φ: X→2X be a set-valued function. Then a ∈ X is a fixed point of φ if

five types of unsupervised learning algorithms that can be cast in the same framework, based on the computation of an embedding for the training points obtained from the principal eigenvectors of a symmetric matrix. Algorithm 1 1. Start from a data set D = {x1 , . . . , xn } with n points in some space. Construct a n × n “neighborhood” or similarity matrix M . Let us denote KD (., .) (or K for shorthand) the two-argument function (sometimes dependent on D) which produces M by Mij = KD (xi , xj ).

et spécifiquement spécialisée dans le domaine des solutions d’ingénierie de centres de données et d’hébergement de systèmes d’information. 1.2 Localisation Groupe SATEC est présent dans six pays : en Espagne, au Portugal à travers sa filiale CONVEX, Consultoria e Integração de Systèmes ; au Maroc grâce à sa part majoritaire dans INTELCOM, S.A. ; en Algérie à travers SATEC Algérie ; en Tunisie à travers SATEC Tunisie et en Angola. Le Groupe SATEC possède des délégations dans les villes suivantes

be exchanged, so we have found the optimum. Solution The solution consists of the last feasible basis variables, here : BV={x4 , x3 , x1 , x7 } The value of these variables is given by : ¯b = {24, 8, 2, 5} respectively. All the other variables are set to 0. The cost function value is therefore : z = −60x1 − 30x2 − 20x3 = −280 In the initial problem, the profit is therefore of 280 euros. We can interpret the solution in the following way : – x1 = 2, so two desks were produced ; – x2 = 0, so zero tables