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Time consistent policy of multiperiod meanvariance problem in stochastic markets
1.  Department of Computing Science, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049 
2.  Department of Computing Science, School of Mathematics and Statistics, Xi'an Jiaotong University, 710049, Xi'an, Shaanxi, China, China 
References:
[1] 
P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, Coherent multiperiod risk adjusted values and Bellman's principle, Annals of Operations Research, 152 (2007), 522. doi: 10.1007/s1047900601326. 
[2] 
S. Basak and G. Chabakauri, Dynamic meanvariance asset allocation, Review of Financial Studies, 23 (2010), 29703016. doi: 10.1093/rfs/hhq028. 
[3] 
D. Bertsimas, G. J. Lauprete and A. Samarov, Shortfall as a risk measure: properties, optimization and app1ications, Journal of Economic Dynamics and Control, 28 (2004), 13531381. 
[4] 
T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, Working Paper, Stockolm School of Economics, 2009. Available from: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1694759. 
[5] 
T. Björk, A. Murgoci and X. Y. Zhou, Mean variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 124. doi: 10.1111/j.14679965.2011.00515.x. 
[6] 
K. Boda and J. A. Filar, Time consistent dynamic risk measures, Mathematical Methods of Operations Research, 63 (2006), 169186. doi: 10.1007/s0018600500451. 
[7] 
M. BrittenJones and A. Neuberger, Option prices, implied price processes, and stochastic volatility, Journal of Finance, 55 (2000), 839866. doi: 10.1111/00221082.00228. 
[8] 
U. Çakmak and S. Özekici, Portfolio optimization in stochastic markets, Mathematical Methods of Operations Research, 63 (2006), 151168. doi: 10.1007/s001860050020x. 
[9] 
E. Çanakoğlu and S. Özekici, Portfolio selection in stochastic markets with exponential utility functions, Annals of Operations Research, 166 (2009), 281297. doi: 10.1007/s1047900804062. 
[10] 
U. Çelikyurt and S. Özekici, Multiperiod portfolio optimization models in stochastic markets using the meanvariance approach, European Journal of Operational Research, 179 (2007), 186202. 
[11] 
Z. P. Chen, G. Li and J. E. Guo, Optimal investment policy in the time consistent meanvariance formulation, Insurance: Mathematics and Economics, 52 (2013), 145156. doi: 10.1016/j.insmatheco.2012.11.007. 
[12] 
Z. P. Chen, G. Li and Y. G. Zhao, Timeconsistent investment policies in Markovian markets: A case of meanvariance analysis, Journal of Economic Dynamic and Control, 40 (2014), 293316. doi: 10.1016/j.jedc.2014.01.011. 
[13] 
Z. P. Chen and J. Liu, Time consistent risk measure under twolevel information structure and its application in dynamic portfolio selection, Working Paper, Xi'an Jiaotong university, 2014. Available from: http://t.cn/RhZ6Vrt. 
[14] 
Z. P. Chen and Y. Wang, Twosided coherent risk measures and their application in realistic portfolio optimization, Journal of Banking and Finance, 32 (2008), 26672673. doi: 10.1016/j.jbankfin.2008.07.004. 
[15] 
Z. P. Chen and L. Yang, Nonlinearly weighted convex risk measure and its application, Journal of Banking and Finance, 35 (2011), 17771793. doi: 10.1016/j.jbankfin.2010.12.004. 
[16] 
P. Cheridito and M. Kupper, Composition of timeconsistent dynamic monetary risk measures in discrete time, Journal of Theoretical and Applied Finance, 14 (2011), 137162. doi: 10.1142/S0219024911006292. 
[17] 
X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic meanvariance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346378. doi: 10.1111/j.14679965.2010.00461.x. 
[18] 
C. Czichowsky, Timeconsistent meanvariance portfolio selection in discrete and continuous time, Finance Stochastic, 17 (2013), 227271. doi: 10.1007/s0078001201899. 
[19] 
R. J. Elliott, T. K. Siu and L. Chan, On pricing barrier options with regime switching, Journal of Computational and Applied Mathematics, 256 (2014), 196210. doi: 10.1016/j.cam.2013.07.034. 
[20] 
L. G. Epstein and S. E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica, 57 (1989), 937969. doi: 10.2307/1913778. 
[21] 
H. Geman and S. Ohana, Timeconsistency in managing a commodity portfolio: A dynamic risk measure approach, Journal of Banking and Finance, 32 (2008), 19912005. doi: 10.1016/j.jbankfin.2007.05.020. 
[22] 
R. Korn and H. Kraft, A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40 (2002), 12501269. doi: 10.1137/S0363012900377791. 
[23] 
D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod meanvariance formulation, Mathematical Finance, 10 (2000), 387406. doi: 10.1111/14679965.00100. 
[24] 
X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic meanvariance portfolio selection with noshorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 15401555. doi: 10.1137/S0363012900378504. 
[25] 
H. J. Lüthi and J. Doege, Convex risk measures for portfolio optimization and concepts of flexibility, Mathematical Programming, 104 (2005), 541559. doi: 10.1007/s101070050628x. 
[26] 
F. Riedel, Dynamic coherent risk measures, Stochastic Processes and their Applications, 112 (2004), 185200. doi: 10.1016/j.spa.2004.03.004. 
[27] 
B. Roorda and J. M. Schumacher, Time consistency conditions for acceptability measures, with an applications to Tail Value at Risk, Insurance: Mathematics and Economics, 40 (2007), 209230. doi: 10.1016/j.insmatheco.2006.04.003. 
[28] 
A. Ruszczyński, Riskaverse dynamic programming for Markov decision processes, Mathematical Programming, Series B, 125 (2010), 235261. doi: 10.1007/s1010701003933. 
[29] 
A. Shapiro, On a time consistency concept in risk averse multistage stochastic programming, Operations Research Letters, 37 (2009), 143147. doi: 10.1016/j.orl.2009.02.005. 
[30] 
M. C. Steinbach, Markowitz revisited: Meanvariance models in financial portfolio analysis, SIAM Review, 43 (2001), 3185. doi: 10.1137/S0036144500376650. 
[31] 
T. Wang, A class of dynamic risk measure, Working Paper, University of British Columbia, 1999. Available from: http://web.cenet.org.cn/upfile/57263.pdf. 
[32] 
J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A timeconsistent strategy, European Journal of Operational Research, 209 (2011), 184201. doi: 10.1016/j.ejor.2010.09.038. 
[33] 
S. Z. Wei and Z. X. Ye, Multiperiod optimization portfolio with bankruptcy control in stochastic market, Applied Mathematics and Computation, 186 (2007), 414425. doi: 10.1016/j.amc.2006.07.108. 
[34] 
L. Xu, R. M. Wang and D. J. Yao, Optimal stochastic investment games under Markov regime switching market, Journal of Industrial and Management Optimization, 10 (2014), 795815. doi: 10.3934/jimo.2014.10.795. 
[35] 
X. Y. Zhou and D. Li, Continuoustime meanvariance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 1933. doi: 10.1007/s002450010003. 
[36] 
S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized meanvariance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447457. doi: 10.1109/TAC.2004.824474. 
show all references
References:
[1] 
P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, Coherent multiperiod risk adjusted values and Bellman's principle, Annals of Operations Research, 152 (2007), 522. doi: 10.1007/s1047900601326. 
[2] 
S. Basak and G. Chabakauri, Dynamic meanvariance asset allocation, Review of Financial Studies, 23 (2010), 29703016. doi: 10.1093/rfs/hhq028. 
[3] 
D. Bertsimas, G. J. Lauprete and A. Samarov, Shortfall as a risk measure: properties, optimization and app1ications, Journal of Economic Dynamics and Control, 28 (2004), 13531381. 
[4] 
T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, Working Paper, Stockolm School of Economics, 2009. Available from: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1694759. 
[5] 
T. Björk, A. Murgoci and X. Y. Zhou, Mean variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 124. doi: 10.1111/j.14679965.2011.00515.x. 
[6] 
K. Boda and J. A. Filar, Time consistent dynamic risk measures, Mathematical Methods of Operations Research, 63 (2006), 169186. doi: 10.1007/s0018600500451. 
[7] 
M. BrittenJones and A. Neuberger, Option prices, implied price processes, and stochastic volatility, Journal of Finance, 55 (2000), 839866. doi: 10.1111/00221082.00228. 
[8] 
U. Çakmak and S. Özekici, Portfolio optimization in stochastic markets, Mathematical Methods of Operations Research, 63 (2006), 151168. doi: 10.1007/s001860050020x. 
[9] 
E. Çanakoğlu and S. Özekici, Portfolio selection in stochastic markets with exponential utility functions, Annals of Operations Research, 166 (2009), 281297. doi: 10.1007/s1047900804062. 
[10] 
U. Çelikyurt and S. Özekici, Multiperiod portfolio optimization models in stochastic markets using the meanvariance approach, European Journal of Operational Research, 179 (2007), 186202. 
[11] 
Z. P. Chen, G. Li and J. E. Guo, Optimal investment policy in the time consistent meanvariance formulation, Insurance: Mathematics and Economics, 52 (2013), 145156. doi: 10.1016/j.insmatheco.2012.11.007. 
[12] 
Z. P. Chen, G. Li and Y. G. Zhao, Timeconsistent investment policies in Markovian markets: A case of meanvariance analysis, Journal of Economic Dynamic and Control, 40 (2014), 293316. doi: 10.1016/j.jedc.2014.01.011. 
[13] 
Z. P. Chen and J. Liu, Time consistent risk measure under twolevel information structure and its application in dynamic portfolio selection, Working Paper, Xi'an Jiaotong university, 2014. Available from: http://t.cn/RhZ6Vrt. 
[14] 
Z. P. Chen and Y. Wang, Twosided coherent risk measures and their application in realistic portfolio optimization, Journal of Banking and Finance, 32 (2008), 26672673. doi: 10.1016/j.jbankfin.2008.07.004. 
[15] 
Z. P. Chen and L. Yang, Nonlinearly weighted convex risk measure and its application, Journal of Banking and Finance, 35 (2011), 17771793. doi: 10.1016/j.jbankfin.2010.12.004. 
[16] 
P. Cheridito and M. Kupper, Composition of timeconsistent dynamic monetary risk measures in discrete time, Journal of Theoretical and Applied Finance, 14 (2011), 137162. doi: 10.1142/S0219024911006292. 
[17] 
X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic meanvariance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346378. doi: 10.1111/j.14679965.2010.00461.x. 
[18] 
C. Czichowsky, Timeconsistent meanvariance portfolio selection in discrete and continuous time, Finance Stochastic, 17 (2013), 227271. doi: 10.1007/s0078001201899. 
[19] 
R. J. Elliott, T. K. Siu and L. Chan, On pricing barrier options with regime switching, Journal of Computational and Applied Mathematics, 256 (2014), 196210. doi: 10.1016/j.cam.2013.07.034. 
[20] 
L. G. Epstein and S. E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica, 57 (1989), 937969. doi: 10.2307/1913778. 
[21] 
H. Geman and S. Ohana, Timeconsistency in managing a commodity portfolio: A dynamic risk measure approach, Journal of Banking and Finance, 32 (2008), 19912005. doi: 10.1016/j.jbankfin.2007.05.020. 
[22] 
R. Korn and H. Kraft, A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40 (2002), 12501269. doi: 10.1137/S0363012900377791. 
[23] 
D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod meanvariance formulation, Mathematical Finance, 10 (2000), 387406. doi: 10.1111/14679965.00100. 
[24] 
X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic meanvariance portfolio selection with noshorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 15401555. doi: 10.1137/S0363012900378504. 
[25] 
H. J. Lüthi and J. Doege, Convex risk measures for portfolio optimization and concepts of flexibility, Mathematical Programming, 104 (2005), 541559. doi: 10.1007/s101070050628x. 
[26] 
F. Riedel, Dynamic coherent risk measures, Stochastic Processes and their Applications, 112 (2004), 185200. doi: 10.1016/j.spa.2004.03.004. 
[27] 
B. Roorda and J. M. Schumacher, Time consistency conditions for acceptability measures, with an applications to Tail Value at Risk, Insurance: Mathematics and Economics, 40 (2007), 209230. doi: 10.1016/j.insmatheco.2006.04.003. 
[28] 
A. Ruszczyński, Riskaverse dynamic programming for Markov decision processes, Mathematical Programming, Series B, 125 (2010), 235261. doi: 10.1007/s1010701003933. 
[29] 
A. Shapiro, On a time consistency concept in risk averse multistage stochastic programming, Operations Research Letters, 37 (2009), 143147. doi: 10.1016/j.orl.2009.02.005. 
[30] 
M. C. Steinbach, Markowitz revisited: Meanvariance models in financial portfolio analysis, SIAM Review, 43 (2001), 3185. doi: 10.1137/S0036144500376650. 
[31] 
T. Wang, A class of dynamic risk measure, Working Paper, University of British Columbia, 1999. Available from: http://web.cenet.org.cn/upfile/57263.pdf. 
[32] 
J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A timeconsistent strategy, European Journal of Operational Research, 209 (2011), 184201. doi: 10.1016/j.ejor.2010.09.038. 
[33] 
S. Z. Wei and Z. X. Ye, Multiperiod optimization portfolio with bankruptcy control in stochastic market, Applied Mathematics and Computation, 186 (2007), 414425. doi: 10.1016/j.amc.2006.07.108. 
[34] 
L. Xu, R. M. Wang and D. J. Yao, Optimal stochastic investment games under Markov regime switching market, Journal of Industrial and Management Optimization, 10 (2014), 795815. doi: 10.3934/jimo.2014.10.795. 
[35] 
X. Y. Zhou and D. Li, Continuoustime meanvariance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 1933. doi: 10.1007/s002450010003. 
[36] 
S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized meanvariance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447457. doi: 10.1109/TAC.2004.824474. 
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