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

• Published in: Journal of global optimization Vol. 68; no. 1; pp. 47 - 69
• Main Authors: Feinstein, Zachary, Rudloff, Birgit
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2017; Springer; Springer Nature B.V
• Subjects: Algorithms; Computer Science; Constraint modelling; Dynamic programming; Markets; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Multivariate analysis; Operations Research/Decision Theory; Optimization; Optimization techniques; Random variables; Real Functions; Risk; Risk assessment; Studies; Time measurement; Bellman’s principle; Dynamic risk measures; 90C39; Dynamic programming; Set-valued risk measures; Transaction costs; 91B30; 46N10; Vector optimization; 26E25
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-016-0459-8

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.