Journal of Global Optimization

, Volume 68, Issue 1, pp 47–69 | Cite as

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

  • Zachary Feinstein
  • Birgit Rudloff


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.


Dynamic risk measures Transaction costs Set-valued risk measures Vector optimization Dynamic programming Bellman’s principle 

Mathematics Subject Classification

91B30 46N10 26E25 90C39 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Electrical and Systems EngineeringWashington University in St. LouisMOUSA
  2. 2.Vienna University of Economics and BusinessInstitute for Statistics and MathematicsViennaAustria

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