1,536 research outputs found
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
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
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