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January  2017, 2: 3 doi: 10.1186/s41546-017-0012-9

A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective

Tomasz R. Bielecki 1,, , Igor Cialenco 1,, and  Marcin Pitera 2,,

1 Department of Applied Mathematics, Illinois Institute of Technology, 10 W 32 nd Str, Building E1, Room 208, Chicago, IL 60616, USA;

2 Institute of Mathematics, Jagiellonian University, Cracow, Poland

Received  March 30, 2016 Revised  December 18, 2016 Published  June 2017

In this work we give a comprehensive overview of the time consistency property of dynamic risk and performance measures, focusing on a the discrete time setup. The two key operational concepts used throughout are the notion of the LMmeasure and the notion of the update rule that, we believe, are the key tools for studying time consistency in a unified framework.
Keywords: Time consistency, Update rule, Dynamic LM-measure, Dynamic risk measure, Dynamic acceptability index, Measure of performance.
Mathematics Subject Classification: 91B30;62P05;97M30;91B06.
Citation: Tomasz R. Bielecki, Igor Cialenco, Marcin Pitera. A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 3-. doi: 10.1186/s41546-017-0012-9
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Bielecki, TR, Cialenco, I, Iyigunler, I, Rodriguez, R:Dynamic Conic Finance:Pricing and hedging via dynamic coherent acceptability indices with transaction costs. Int. J. Theor. Appl. Finance 16(01), 1350002 (2013) Google Scholar

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[15]

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[16]

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[19]

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[24]

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[31]

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Cherny, A:Capital allocation and risk contribution with discrete-time coherent risk. Math. Finance 19(1), 13-40 (2009) Google Scholar

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Cherny, A:Risk-reward optimization with discrete-time coherent risk. Math. Finance 20(4), 571-595(2010) Google Scholar

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Cherny, A, Madan, DB:Pricing and hedging in incomplete markets with coherent risk, 21 (2006). https://ssrn.com/abstract=904806 Google Scholar

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Cheridito, P, Delbaen, F, Kupper, M:Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab 11(3), 57-106 (2006) Google Scholar

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Cheridito, P, Stadje, M:Time-inconsistency of VaR and time-consistent alternatives. Finance Res. Lett 6(1), 40-46 (2009) Google Scholar

[40]

Cohen, SN, Elliott, RJ:A general theory of finite state backward stochastic difference equations.Stochastic Process. Appl 120(4), 442-466 (2010) Google Scholar

[41]

Cohen, SN, Elliott, RJ:Backward stochastic difference equations and nearly-time-consistent nonlinear expectations. SIAM J. Control Optim 49(1), 125-139 (2011) Google Scholar

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Coquet, F, Hu, Y, Mémin, J, Peng, S:Filtration-consistent nonlinear expectations and related g-xpectations. Probab. Theory Relat. Fields 123(1), 1-27 (2002) Google Scholar

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