57 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
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
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