50 research outputs found
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 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
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 with image space in the power set of . 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
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
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
Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems
Two approximation algorithms for solving convex vector optimization problems
(CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual
problem simultaneously. The first algorithm is an extension of Benson's outer
approximation algorithm, and the second one is a dual variant of it. Both
algorithms provide an inner as well as an outer approximation of the (upper and
lower) images. Only one scalar convex program has to be solved in each
iteration. We allow objective and constraint functions that are not necessarily
differentiable, allow solid pointed polyhedral ordering cones, and relate the
approximations to an appropriate \epsilon-solution concept. Numerical examples
are provided