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

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Abstract

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.

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Authors and Affiliations

  1. Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO, USA

    Zachary Feinstein

  2. Vienna University of Economics and Business, Institute for Statistics and Mathematics, Vienna, Austria

    Birgit Rudloff

Authors
  1. Zachary Feinstein
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  2. Birgit Rudloff
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Correspondence to Zachary Feinstein.

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Feinstein, Z., Rudloff, B. A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle. J Glob Optim 68, 47–69 (2017). https://doi.org/10.1007/s10898-016-0459-8

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  • DOI: https://doi.org/10.1007/s10898-016-0459-8

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