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Pontryagin's principle for local solutions of optimal control governed by the 2D NavierStokes equations with mixed controlstate constraints
Optimal trendfollowing trading rules under a threestate regime switching model
1.  Department of Mathematics and Actuarial Science, Roosevelt University, Chicago, IL 60605, United States 
2.  Department of Mathematics, University of Georgia, Athens, GA 30602 
References:
[1] 
N. P. B. Bollen, Valuing options in regimeswitching models, Journal of Derivatives, 6 (1998), 3849. doi: 10.3905/jod.1998.408011. 
[2] 
J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497514. doi: 10.1142/S0219024902001523. 
[3] 
A. Cadenillas and S. R. Pliska, Optimal trading of a security when there are taxes and transaction costs, Finance & Stochastics, 3 (1999), 137165. doi: 10.1007/s007800050055. 
[4] 
G. M. Constantinides, Capital market equilibrium with personal tax, Econometrica, 51 (1983), 611636. doi: 10.2307/1912150. 
[5] 
M. Dai, Q. Zhang and Q. J. Zhu, Trend following trading under a regime switching model, SIAM J. Financial Math, 1 (2010), 780810. doi: 10.1137/090770552. 
[6] 
R. M. Dammon and C. S. Spatt, The optimal trading and pricing of securities with asymmetric capital gains taxes and transaction costs, Rev. Financial Studies, 9 (1996), 921952. doi: 10.1093/rfs/9.3.921. 
[7] 
R. J. Elliott, "Stochastic Calculus and Applications," SpringerVerlag, New York, 1982. 
[8] 
X. Guo, "Inside Information and Stock Fluctuations," Ph.D thesis, Rutgers University, 1999. 
[9] 
X. Guo and Q. Zhang, Optimal selling rules in a regime switching model, IEEE Trans. Automatic Control, 50 (2005), 14501455. doi: 10.1109/TAC.2005.854657. 
[10] 
J. D. Hamilton, A new approach to the economic analysis of nonstationary time series, Econometrica, 57 (1989), 357384. doi: 10.2307/1912559. 
[11] 
K. Helmes, Computing optimal selling rules for stocks using linear programming, in "Mathematics of Finance" (eds. G. Yin and Q. Zhang), Contemporary Mathematics, 351, American Mathematical Society, Providence, RI, (2004), 187198. 
[12] 
T. C. Johnson and M. Zervos, The optimal timing of investment decisions, work in progress, 2006. 
[13] 
I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Springer, New York, 1998. 
[14] 
H. T. Kong and Q. Zhang, An optimal trading rule of a meanreverting asset, Discrete and Continuous Dynamical System Series B, 14 (2010), 14031417. doi: 10.3934/dcdsb.2010.14.1403. 
[15] 
H. T. Kong, Q. Zhang and G. Yin, A trendfollowing strategy: Conditions for optimality, Automatica, 47 (2011), 661667. doi: 10.1016/j.automatica.2011.01.039. 
[16] 
R. Liu, G. Yin and Q. Zhang, Option pricing in a regime switching model using the fast Fourier transform, Applied Mathematics and Stochastic Analysis, 2006, Art. ID 18109, 22 pp. 
[17] 
A. Løkka and M. Zervos, Longterm optimal real investment strategies in the presence of adjustment costs, work in progress, 2007. 
[18] 
G. B. Di Masi, Y. M. Kabanov and W. J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory of Probability and Applications, 39 (1994), 173181. doi: 10.1137/1139008. 
[19] 
A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control Optim., 46 (2007), 839876. doi: 10.1137/050640758. 
[20] 
B. Øksendal, "Stochastic Differential Equations," 6^{th} edition, SpringerVerlag, New York, 2003. 
[21] 
D. D. Yao, Q. Zhang and X. Y. Zhou, A regimeswitching model for European options, in "Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems" (eds. H. Yan, G. Yin and Q. Zhang), International Series in Operations Research and Management Sciences, 94, Springer, New York, (2006), 281300. 
[22] 
G. Yin, R. H. Liu and Q. Zhang, Recursive algorithms for stock liquidation: A stochastic optimization approach, SIAM J. Optim., 13 (2002), 240263. doi: 10.1137/S1052623401392901. 
[23] 
G. Yin and C. Zhu, "Hybrid Switching Diffusions: Properties and Applications," Springer, New York, 2010. 
[24] 
H. Zhang and Q. Zhang, Trading a meanreverting asset: Buy low and sell high, Automatica J. IFAC, 44 (2008), 15111518. doi: 10.1016/j.automatica.2007.11.003. 
[25] 
Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 6487. doi: 10.1137/S0363012999356325. 
[26] 
Q. Zhang and G. Yin, Nearly optimal asset allocation in hybrid stockinvestment models, J. Optim. Theory Appl., 121 (2004), 419444. doi: 10.1023/B:JOTA.0000037412.23243.6c. 
[27] 
X. Y. Zhou and G. Yin, Markowitz's meanvariance portfolio selection with regime switching: A continuoustime model, SIAM J. Control Optim., 42 (2003), 14661482. doi: 10.1137/S0363012902405583. 
show all references
References:
[1] 
N. P. B. Bollen, Valuing options in regimeswitching models, Journal of Derivatives, 6 (1998), 3849. doi: 10.3905/jod.1998.408011. 
[2] 
J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497514. doi: 10.1142/S0219024902001523. 
[3] 
A. Cadenillas and S. R. Pliska, Optimal trading of a security when there are taxes and transaction costs, Finance & Stochastics, 3 (1999), 137165. doi: 10.1007/s007800050055. 
[4] 
G. M. Constantinides, Capital market equilibrium with personal tax, Econometrica, 51 (1983), 611636. doi: 10.2307/1912150. 
[5] 
M. Dai, Q. Zhang and Q. J. Zhu, Trend following trading under a regime switching model, SIAM J. Financial Math, 1 (2010), 780810. doi: 10.1137/090770552. 
[6] 
R. M. Dammon and C. S. Spatt, The optimal trading and pricing of securities with asymmetric capital gains taxes and transaction costs, Rev. Financial Studies, 9 (1996), 921952. doi: 10.1093/rfs/9.3.921. 
[7] 
R. J. Elliott, "Stochastic Calculus and Applications," SpringerVerlag, New York, 1982. 
[8] 
X. Guo, "Inside Information and Stock Fluctuations," Ph.D thesis, Rutgers University, 1999. 
[9] 
X. Guo and Q. Zhang, Optimal selling rules in a regime switching model, IEEE Trans. Automatic Control, 50 (2005), 14501455. doi: 10.1109/TAC.2005.854657. 
[10] 
J. D. Hamilton, A new approach to the economic analysis of nonstationary time series, Econometrica, 57 (1989), 357384. doi: 10.2307/1912559. 
[11] 
K. Helmes, Computing optimal selling rules for stocks using linear programming, in "Mathematics of Finance" (eds. G. Yin and Q. Zhang), Contemporary Mathematics, 351, American Mathematical Society, Providence, RI, (2004), 187198. 
[12] 
T. C. Johnson and M. Zervos, The optimal timing of investment decisions, work in progress, 2006. 
[13] 
I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Springer, New York, 1998. 
[14] 
H. T. Kong and Q. Zhang, An optimal trading rule of a meanreverting asset, Discrete and Continuous Dynamical System Series B, 14 (2010), 14031417. doi: 10.3934/dcdsb.2010.14.1403. 
[15] 
H. T. Kong, Q. Zhang and G. Yin, A trendfollowing strategy: Conditions for optimality, Automatica, 47 (2011), 661667. doi: 10.1016/j.automatica.2011.01.039. 
[16] 
R. Liu, G. Yin and Q. Zhang, Option pricing in a regime switching model using the fast Fourier transform, Applied Mathematics and Stochastic Analysis, 2006, Art. ID 18109, 22 pp. 
[17] 
A. Løkka and M. Zervos, Longterm optimal real investment strategies in the presence of adjustment costs, work in progress, 2007. 
[18] 
G. B. Di Masi, Y. M. Kabanov and W. J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory of Probability and Applications, 39 (1994), 173181. doi: 10.1137/1139008. 
[19] 
A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control Optim., 46 (2007), 839876. doi: 10.1137/050640758. 
[20] 
B. Øksendal, "Stochastic Differential Equations," 6^{th} edition, SpringerVerlag, New York, 2003. 
[21] 
D. D. Yao, Q. Zhang and X. Y. Zhou, A regimeswitching model for European options, in "Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems" (eds. H. Yan, G. Yin and Q. Zhang), International Series in Operations Research and Management Sciences, 94, Springer, New York, (2006), 281300. 
[22] 
G. Yin, R. H. Liu and Q. Zhang, Recursive algorithms for stock liquidation: A stochastic optimization approach, SIAM J. Optim., 13 (2002), 240263. doi: 10.1137/S1052623401392901. 
[23] 
G. Yin and C. Zhu, "Hybrid Switching Diffusions: Properties and Applications," Springer, New York, 2010. 
[24] 
H. Zhang and Q. Zhang, Trading a meanreverting asset: Buy low and sell high, Automatica J. IFAC, 44 (2008), 15111518. doi: 10.1016/j.automatica.2007.11.003. 
[25] 
Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40 (2001), 6487. doi: 10.1137/S0363012999356325. 
[26] 
Q. Zhang and G. Yin, Nearly optimal asset allocation in hybrid stockinvestment models, J. Optim. Theory Appl., 121 (2004), 419444. doi: 10.1023/B:JOTA.0000037412.23243.6c. 
[27] 
X. Y. Zhou and G. Yin, Markowitz's meanvariance portfolio selection with regime switching: A continuoustime model, SIAM J. Control Optim., 42 (2003), 14661482. doi: 10.1137/S0363012902405583. 
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