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

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.

25 pages, 5 figures

References in corpus (5)

• Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems
• Benson type algorithms for linear vector optimization and applications
• Multiportfolio time consistency for set-valued convex and coherent risk measures
• Set-valued shortfall and divergence risk measures
• An Algorithm to Solve Polyhedral Convex Set Optimization Problems

Cited by in corpus (3)

• Markov Decision Processes with Risk-Sensitive Criteria: An Overview
• Set-Valued Risk Measures as Backward Stochastic Difference Inclusions and Equations
• Deep Learning the Efficient Frontier of Convex Vector Optimization Problems