Testing composite hypotheses via convex duality

We study the problem of testing composite hypotheses versus composite alternatives, using a convex duality approach. In contrast to classical results obtained by Krafft and Witting (Z. Wahrsch. Verw. Gebiete 7 (1967) 289--302), where sufficient optimality conditions are derived via Lagrange duality, we obtain necessary and sufficient optimality conditions via Fenchel duality under compactness assumptions. This approach also differs from the methodology developed in Cvitani\'{c} and Karatzas (Bernoulli 7 (2001) 79--97).Comment: Published in at http://dx.doi.org/10.3150/10-BEJ249 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.Comment: 25 pages, 5 figure

On the Dual of the Solvency Cone

A solvency cone is a polyhedral convex cone which is used in Mathematical Finance to model proportional transaction costs. It consists of those portfolios which can be traded into nonnegative positions. In this note, we provide a characterization of its dual cone in terms of extreme directions and discuss some consequences, among them: (i) an algorithm to construct extreme directions of the dual cone when a corresponding "contribution scheme" is given; (ii) estimates for the number of extreme directions; (iii) an explicit representation of the dual cone for special cases. The validation of the algorithm is based on the following easy-to-state but difficult-to-solve result on bipartite graphs: Running over all spanning trees of a bipartite graph, the number of left degree sequences equals the number of right degree sequences.Comment: 15 page

Time consistency of dynamic risk measures in markets with transaction costs

The paper concerns primal and dual representations as well as time consistency of set-valued dynamic risk measures. Set-valued risk measures appear naturally when markets with transaction costs are considered and capital requirements can be made in a basket of currencies or assets. Time consistency of scalar risk measures can be generalized to set-valued risk measures in different ways. The most intuitive generalization is called time consistency. We will show that the equivalence between a recursive form of the risk measure and time consistency, which is a central result in the scalar case, does not hold in the set-valued framework. Instead, we propose an alternative generalization, which we will call multi-portfolio time consistency and show in the main result of the paper that this property is indeed equivalent to the recursive form as well as to an additive property for the acceptance sets. Multi-portfolio time consistency is a stronger property than time consistency. In the scalar case, both notions coincide

Multiportfolio time consistency for set-valued convex and coherent risk measures

Equivalent characterizations of multiportfolio time consistency are deduced for closed convex and coherent set-valued risk measures on $L^p(\Omega,\mathcal F, P; R^d)$ with image space in the power set of $L^p(\Omega,\mathcal F_t,P;R^d)$. In the convex case, multiportfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multiportfolio time consistency is equivalent to a generalized version of stability of the dual variables. As examples, the set-valued entropic risk measure with constant risk aversion coefficient is shown to satisfy the cocycle condition for its minimal penalty functions, the set of superhedging portfolios in markets with proportional transaction costs is shown to have the stability property and in markets with convex transaction costs is shown to satisfy the composed cocycle condition, and a multiportfolio time consistent version of the set-valued average value at risk, the composed AV@R, is given and its dual representation deduced

A Supermartingale Relation for Multivariate Risk Measures

The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on multivariate risks.Comment: 40 page

A Parametric Simplex Algorithm for Linear Vector Optimization Problems

In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (Evans-Steuer) algorithm [12]. Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Loehne [16], that is, it finds a subset of efficient solutions that allows to generate the whole frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczynski and Vanderbei [21]. The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems. Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm (Benson algorithm in [13]), that also provides a solution in the sense of [16], and with Evans-Steuer algorithm [12]. The results show that for non-degenerate problems the proposed algorithm outperforms Benson algorithm and is on par with Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [13] excels the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than Evans-Steuer algorithm.Comment: 27 pages, 4 figures, 5 table

What makes countries initiate WTO disputes on food-related measures?

This paper analyses relevant parameters for initiating a World Trade Organization (WTO) dispute on foodrelated measures and thereby contributes to the question how open de facto the system is for different types of countries. The empirical analysis differs from existing assessments by focussing on agri-food related disputes, thereby allowing for a more in-depth analysis of specific country characteristics not considered in previous studies. The results show that some determinants such as legal capacity and monetary means are not statistically significant for agri-food dispute initiations. This is the case for own protectionist behavior and endured protectionism which lower and enlarge the probability to complain, respectively.WTO disputes, agri-food disputes, binomial distribution model of complaints, International Relations/Trade,

The Bioterrorism Act of the USA and international food trade: evaluating WTO conformity and effects on bilateral imports

The September 11th event focused the world’s attention on the threat of bioterrorism on the food chain. As a consequence, the USA implemented the Bioterrorism Act (BTA) containing new import requirements that can be classified as non-tariff barriers (NTBs). This paper analyses these NTBs by performing an assessment of WTO conformity and trade impact: hereby general problems in the analysis of bioterrorist risks are explored as for this new and unknown threat explicit WTO rules are still missing. Additionally, in exploring the BTA relevant process standard rules laid out by the WTO, the analysis indicates the extensive flexibility provided in this framework. This leads to larger scope for national polices on process standards compared to product standards (e.g. residua levels). The empirical trade flow analysis illustrates differences in the compliance costs between countries. This differentiation can be caused by learning costs that may differ among countries. The analysis highlights that perishable products and countries with small import quantities are mostly affected.food terrorism, non-tariff barriers, trade facilitation, Bioterrorism Act, international food trade, SPS Agreement, International Relations/Trade,