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

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

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.
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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References:
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[13]

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

[14]

Bielecki, TR, Cialenco, I, Pitera, M:A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time. Preprint (2014a) Google Scholar

[15]

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[16]

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[17]

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[18]

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[19]

Bion-Nadal, J:Dynamic risk measures:time consistency and risk measures from BMO martingales.Finance Stoch 12(2), 219-244 (2008) Google Scholar

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[22]

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[23]

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[24]

Bion-Nadal, J, Kervarec, M:Dynamic risk measuring under model uncertainty:taking advantage of the hidden probability measure. Preprint (2010) Google Scholar

[25]

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[26]

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[27]

Cheridito, P, Kupper, M:Recursiveness of indifference prices and translation-invariant preferences. Math.Financ. Econ 2(3), 173-188 (2009) Google Scholar

[28]

Cheridito, P, Kupper, M:Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Finance 14(1), 137-162 (2011) Google Scholar

[29]

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[30]

Cheridito, P, Li, T:Risk measures on Orlicz hearts. Math. Finance 19(2), 189-214 (2009) Google Scholar

[31]

Cherny, A:Weighted V@R and its properties. Finance Stoch 10(3), 367-393 (2006) Google Scholar

[32]

Cherny, A:Pricing and hedging European options with discrete-time coherent risk. Finance Stoch 11(4), 537-569 (2007) Google Scholar

[33]

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

[34]

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

[36]

Cherny, A, Madan, DB:New measures for performance evaluation. Rev. Financial Stud 22(7), 2571-2606(2009) Google Scholar

[37]

Cherny, A, Madan, DB:Markets as a counterparty:An introduction to conic finance. Int. J. Theor. Appl.Finance (IJTAF) 13(08), 1149-1177 (2010) Google Scholar

[38]

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

[39]

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

[42]

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

[43]

Davis, M, Lleo, S:Risk-Sensitive Investment Management, volume 19 of Advanced Series on Statistical Science & Applied Probability. World Sci (2014) Google Scholar

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Delbaen, F:Coherent risk measures. Scuola Normale Superiore (2000) Google Scholar

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Delbaen, F:Coherent risk measures on general probability spaces. Advances in finance and stochastics, pp. 1-37. Springer (2002) Google Scholar

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