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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)

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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)

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Bielecki, TR, Cialenco, I, Zhang, Z:Dynamic coherent acceptability indices and their applications to finance. Math. Finance 24(3), 411-441 (2014b)

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[1]

Acciaio, B, Föllmer, H, Penner, I:Risk assessment for uncertain cash flows:model ambiguity, discounting ambiguity, and the role of bubbles. Finance Stochastics 16, 669-709 (2012)

[2]

Acciaio, B, Penner, I. In:Di Nunno, G, Øksendal, B (eds.):Dynamic risk measures, pp. 1-34. Springer(2011)

[3]

Artzner, P, Delbaen, F, Eber, JM, Heath, D:Coherent measures of risk. Math. Finance 9(3), 203-228(1999)

[4]

Artzner, P, Delbaen, F, Eber, JM, Heath, D, Ku, H:Coherent multiperiod risk measurement. Preprint(2002a)

[5]

Artzner, P, Delbaen, F, Eber, JM, Heath, D, Ku, H:Multiperiod risk and coherent multiperiod risk measurement. Mimeo. ETH ZLurich, Switzerland (2002b)

[6]

Artzner, P, Delbaen, F, Eber, JM, Heath, D, Ku, H:Coherent multiperiod risk adjusted values and Bellman's principle. Ann. Oper. Res 152, 5-22 (2007)

[7]

Barron, EN, Cardaliaguet, P, Jensen, R:Conditional essential suprema with applications. Appl. Math.Optim 48(3), 229-253 (2003)

[8]

Barrieu, P, El Karoui, N:Optimal derivatives design under dynamic risk measures. Mathematics of finance, volume 351 of Contemp. Math, pp. 13-25 (2004). Amer. Math. Soc Barrieu, P, El Karoui, N:Inf-convolution of risk measures and optimal risk transfer. Finance Stoch 9(2), 269-298 (2005)

[9]

Barrieu, P, El Karoui, N:Pricing, hedging and optimally designing derivatives via minimization of risk measures. In:Carmona, R (ed.) Indifference Pricing. Princeton University Press (2007)

[10]

Bellman, RE, Dreyfus, SE:Applied dynamic programming. Princeton University Press (1962)

[11]

Biagini, S, Bion-Nadal, J:Dynamic quasi-concave performance measures. J. Math. Econ 55, 143-153(2014)

[12]

Bielecki, TR, Pliska, SR:Economic properties of the risk sensitive criterion for portfolio management.Rev. Account. Finance 2, 3-17 (2003)

[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)

[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)

[15]

Bielecki, TR, Cialenco, I, Zhang, Z:Dynamic coherent acceptability indices and their applications to finance. Math. Finance 24(3), 411-441 (2014b)

[16]

Bielecki, TR, Cialenco, I, Pitera, M:Dynamic limit growth indices in discrete time. Stochastic Models 31, 494-523 (2015a)

[17]

Bielecki, TR, Cialenco, I, Chen, T:Dynamic conic finance via Backward Stochastic Difference Equations.SIAM J. Finan. Math 6(1), 1068-1122 (2015b)

[18]

Bielecki, TR, Cialenco, I, Drapeau, S, Karliczek, M:Dynamic assessment indices. Stochastics:Int. J.Probab. Stochastic Process 88(1), 1-44 (2016)

[19]

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

[20]

Bion-Nadal, J:Bid-ask dynamic pricing in financial markets with transaction costs and liquidity risk. J.Math. Econ 45(11), 738-750 (2009a)

[21]

Bion-Nadal, J:Time consistent dynamic risk processes. Stochastic Process. Appl 119(2), 633-654 (2009b)

[22]

Bion-Nadal, J:Dynamic risk measures and path-dependent second order PDEs. In:Espen Benth, F, Di Nunno, G (eds.) Stochastics of Environmental and Financial Economics, volume 138 of Springer Proceedings in Mathematics & Statistics, pp. 147-178. Springer (2016)

[23]

Boda, K, Filar, J:Time consistent dynamic risk measures. Math. Methods Oper. Res 63(1), 169-186 (2006)

[24]

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

[25]

Bion-Nadal, J, Kervarec, M:Risk measuring under model uncertainty. Ann. App. Prob 1, 213-238 (2012)

[26]

Carpentier, P, Chancelier, JP, Cohen, G, De Lara, M, Girardeau, P:Dynamic consistency for stochastic optimal control problems. Ann. Oper. Res 200(1), 247-263 (2012)

[27]

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

[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)

[29]

Cheridito, P, Li, T:Dual characterization of properties of risk measures on Orlicz hearts. Math. Financ.Econ 2(1), 29-55 (2008)

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[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)

[38]

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

[39]

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

[40]

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

[41]

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

[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)

[43]

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

[44]

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

[45]

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

[46]

Delbaen, F:The structure of m-stable sets and in particular of the set of risk neutral measures. In memoriam Paul-André Meyer:Séminaire de Probabilités XXXIX, volume 1874 of Lecture Notes in Math., pp. 215-258. Springer, Berlin (2006)

[47]

Delbaen, F:Monetary Utility Functions, volume 3 of CSFI Lecture Notes Series. Osaka University Press(2012)

[48]

Delbaen, F, Peng, S, Rosazza Gianin, E:Representation of the penalty term of dynamic concave utilities.Finance Stoch 14, 449-472 (2010)

[49]

Detlefsen, K, Scandolo, G:Conditional and dynamic convex risk measures. Finance Stochastics 9(4), 539-561 (2005)

[50]

Drapeau, S:Risk Preferences and their Robust Representation. PhD thesis, Humboldt Universität zu Berlin (2010)

[51]

Drapeau, S, Kupper, M:Risk preferences and their robust representation. Math. Oper. Res 38(1), 28-62(2013)

[52]

Elliott, RJ, Siu, TK, Cohen, SN:Backward stochastic difference equations for dynamic convex risk measures on a binomial tree. J. Appl. Probab 52(3), 2015 (2015)

[53]

Epstein, LG, Schneider, M:Recursive multiple-priors. J. Econom. Theory 113(1), 1-31 (2003)

[54]

Fan, J, Ruszczyński, A:Process-based risk measures for observable and partially observable discrete-time controlled systems. Preprint (2014)

[55]

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