1,439 research outputs found
Rank properties of exposed positive maps
Let \cK and \cH be finite dimensional Hilbert spaces and let \fP denote
the cone of all positive linear maps acting from \fB(\cK) into \fB(\cH). We
show that each map of the form or is an
exposed point of \fP. We also show that if a map is an exposed point
of \fP then either is rank 1 non-increasing or \rank\phi(P)>1 for
any one-dimensional projection P\in\fB(\cK).Comment: 6 pages, last section removed - it will be a part of another pape
On the dual representation of coherent risk measures
A classical result in risk measure theory states that every coherent risk measure has a dual representation as the supremum of certain expected value over a risk envelope. We study this topic in more detail. The related issues include: (1) Set operations of risk envelopes and how they change the risk measures, (2) The structure of risk envelopes of popular risk measures, (3) Aversity of risk measures and its impact to risk envelopes, and (4) A connection between risk measures in stochastic optimization and uncertainty sets in robust optimization
Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging
The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation
Linear -positive sets and their polar subspaces
In this paper, we define a Banach SNL space to be a Banach space with a
certain kind of linear map from it into its dual, and we develop the theory of
linear -positive subsets of Banach SNL spaces with Banach SNL dual spaces.
We use this theory to give simplified proofs of some recent results of
Bauschke, Borwein, Wang and Yao, and also of the classical Brezis-Browder
theorem.Comment: 11 pages. Notational changes since version
Local Volatility Calibration by Optimal Transport
The calibration of volatility models from observable option prices is a
fundamental problem in quantitative finance. The most common approach among
industry practitioners is based on the celebrated Dupire's formula [6], which
requires the knowledge of vanilla option prices for a continuum of strikes and
maturities that can only be obtained via some form of price interpolation. In
this paper, we propose a new local volatility calibration technique using the
theory of optimal transport. We formulate a time continuous martingale optimal
transport problem, which seeks a martingale diffusion process that matches the
known densities of an asset price at two different dates, while minimizing a
chosen cost function. Inspired by the seminal work of Benamou and Brenier [1],
we formulate the problem as a convex optimization problem, derive its dual
formulation, and solve it numerically via an augmented Lagrangian method and
the alternative direction method of multipliers (ADMM) algorithm. The solution
effectively reconstructs the dynamic of the asset price between the two dates
by recovering the optimal local volatility function, without requiring any time
interpolation of the option prices
Risk scoring models for trade credit in small and medium enterprises
Trade credit refers to providing goods and services on a deferred payment basis. Commercial credit management is a matter of great importance for most small and medium enterprises (SMEs), since it represents a significant portion of their assets. Commercial lending involves assuming some credit risk due to exposure to default. Thus, the management of trade credit and payment delays is strongly related to the liquidation and bankruptcy of enterprises. In this paper we study the relationship between trade credit management and the level of risk in SMEs. Despite its relevance for most SMEs, this problem has not been sufficiently analyzed in the existing literature. After a brief review of existing literature, we use a large database of enterprises to analyze data and propose a multivariate decision-tree model which aims at explaining the level of risk as a function of several variables, both of financial and non-financial nature. Decision trees replace the equation in parametric regression models with a set of rules. This feature is an important aid for the decision process of risk experts, as it allows them to reduce time and then the economic cost of their decisions
Activity Identification and Local Linear Convergence of Douglas--Rachford/ADMM under Partial Smoothness
Convex optimization has become ubiquitous in most quantitative disciplines of
science, including variational image processing. Proximal splitting algorithms
are becoming popular to solve such structured convex optimization problems.
Within this class of algorithms, Douglas--Rachford (DR) and alternating
direction method of multipliers (ADMM) are designed to minimize the sum of two
proper lower semi-continuous convex functions whose proximity operators are
easy to compute. The goal of this work is to understand the local convergence
behaviour of DR (resp. ADMM) when the involved functions (resp. their
Legendre-Fenchel conjugates) are moreover partly smooth. More precisely, when
both of the two functions (resp. their conjugates) are partly smooth relative
to their respective manifolds, we show that DR (resp. ADMM) identifies these
manifolds in finite time. Moreover, when these manifolds are affine or linear,
we prove that DR/ADMM is locally linearly convergent. When and are
locally polyhedral, we show that the optimal convergence radius is given in
terms of the cosine of the Friedrichs angle between the tangent spaces of the
identified manifolds. This is illustrated by several concrete examples and
supported by numerical experiments.Comment: 17 pages, 1 figure, published in the proceedings of the Fifth
International Conference on Scale Space and Variational Methods in Computer
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