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Martingale characterizations of risk-averse stochastic optimization problems

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Abstract

This paper addresses risk awareness of stochastic optimization problems. Nested risk measures appear naturally in this context, as they allow beneficial reformulations for algorithmic treatments. The reformulations presented extend usual dynamic equations by involving risk awareness in the problem formulation. Nested risk measures are built on risk measures, which originate by conditioning on the history of a stochastic process. We derive martingale properties of these risk measures and use them to prove continuity. It is demonstrated that stochastic optimization problems, which incorporate risk awareness via nesting risk measures, are continuous with respect to the natural distance governing these optimization problems, the nested distance.

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Acknowledgements

We would like to thank Prof. Shapiro for proposing to elaborate the continuity relations of nested risk measures with respect to the nested distance.

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  1. Technische Universität Chemnitz, 09126, Chemnitz, Germany

    Alois Pichler & Ruben Schlotter

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  1. Alois Pichler
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Correspondence to Alois Pichler.

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Special Issue Math. Prog. on “The interface between optimization and probability”.

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Pichler, A., Schlotter, R. Martingale characterizations of risk-averse stochastic optimization problems. Math. Program. 181, 377–403 (2020). https://doi.org/10.1007/s10107-019-01391-2

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