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January  2017, 13(1): 23-46. doi: 10.3934/jimo.2016002

Consumption-portfolio optimization and filtering in a hidden Markov-modulated asset price model

1. 

Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

2. 

Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

Received  July 2013 Revised  June 2015 Published  March 2016

We study a consumption-portfolio optimization problem in a hidden Markov-modulated asset price model with multiple risky assets, where model uncertainty is present. Under this modeling framework, the appreciation rates of risky shares are modulated by a continuous-time, finite-state hidden Markov chain whose states represent different modes of the model. We consider the situation where an economic agent only has access to information about the price processes of risky shares and aims to maximize the expected, discounted utility from intermediate consumption and terminal wealth within a finite horizon. The standard innovations approach in filtering theory is then used to transform the partially observed consumption-portfolio optimization problem to the one with complete observations. Robust filters of the chain and estimates of some other parameters are presented. Using the stochastic maximum principle, we derive a closed-form solution of an optimal consumption-portfolio strategy in the case of a power utility.

Citation: Yang Shen, Tak Kuen Siu. Consumption-portfolio optimization and filtering in a hidden Markov-modulated asset price model. Journal of Industrial and Management Optimization, 2017, 13 (1) : 23-46. doi: 10.3934/jimo.2016002
References:
[1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd edition, Springer, New York, 2003. 
[2] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979.  doi: 10.1137/1.9780898719048.
[3]

J. M. C. Clark, The design of robust approximations to the stochastic differential equations for nonlinear filtering, in Communications Systems and Random Process Theory (ed. J. K. Skwirzynski), NATO Advanced Study Inst. Ser., Ser. E: Appl. Sci., No. 25, Sijthoff & Noordhoff, Alphen aan den Rijn, 1978,721-734.

[4] R. J. Elliott, Stochastic Calculus and Applications, Springer, Berlin-Heidelberg-New York, 1982. 
[5] R. J. ElliottL. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer, Berlin-Heidelberg-New York, 1994. 
[6]

R. J. ElliottT. K. Siu and A. Badescu, On mean-variance portfolio selection under a Markovian regime-switching model, Economic Modelling, 27 (2010), 678-686.  doi: 10.1016/j.econmod.2010.01.007.

[7]

R. J. Elliott and T. K. Siu, An HMM approach for optimal investment of an insurer, International Journal of Robust Nonlinear Control, 22 (2012), 778-807.  doi: 10.1002/rnc.1727.

[8] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin-Heidelberg-New York, 1975. 
[9]

S. M. Goldfeld and R. E. Quandt, A Markov model for switching regressions, Journal of Econometrics, 1 (1973), 3-16.  doi: 10.1016/0304-4076(73)90002-X.

[10]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.

[11]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559.

[12] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008. 
[13] M. JeanblancM. Yor and M. Chesney, Mathematical Methods for Financial Markets, Springer, New York, 2009.  doi: 10.1007/978-1-84628-737-4.
[14] G. Kallianpur, Stochastic Filtering Theory, Springer, Berlin-Heidelberg-New York, 1980. 
[15]

R. KornT. K. Siu and A. Zhang, Asset allocation for a DC pension fund under regime-switching environment, European Actuarial Journal, 1 (2011), 361-377.  doi: 10.1007/s13385-011-0021-5.

[16]

H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.

[17]

R. C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time model, The Review of Economics and Statistics, 51 (1969), 247-257. 

[18]

R. C. Merton, Optimal consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.

[19]

R. C. Merton, On estimating the expected return on the market, Journal of Financial Economics, 8 (1980), 323-361.  doi: 10.1016/0304-405X(80)90007-0.

[20]

W. Putschögl and J. Sass, Optimal consumption and investment under partial information, Decisions in Economics and Finance, 31 (2008), 137-170.  doi: 10.1007/s10203-008-0082-3.

[21]

R. E. Quandt, The estimation of parameters of linear regression system obeying two separate regimes, Journal of the American Statistical Association, 55 (1958), 873-880.  doi: 10.1080/01621459.1958.10501484.

[22]

U. Rieder and N. Bäuerle, Portfolio optimization with unobservable Markov-modulated drift process, Journal of Applied Probability, 42 (2005), 362-378.  doi: 10.1239/jap/1118777176.

[23]

J. Sass and U.G. Haussmann, Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain, Finance and Stochastics, 8 (2004), 553-577.  doi: 10.1007/s00780-004-0132-9.

[24]

A. SchiedH. Föllmer and S. Weber, Robust preferences and robust portfolio choice, Handbook of Numerical Analysis, 15 (2009), 29-87.  doi: 10.1016/S1570-8659(08)00002-1.

[25]

Y. Shen and T. K. Siu, Asset allocation under stochastic interest rate with regime switching, Economic Modelling, 29 (2012), 1126-1136.  doi: 10.1016/j.econmod.2012.03.024.

[26]

T. K. Siu, Long-term strategic asset allocation with inflation risk and regime switching, Quantitative Finance, 11 (2011), 1565-1580.  doi: 10.1080/14697680903055588.

[27]

T. K. Siu, A BSDE approach to risk-based asset allocation of pension funds with regime switching, Annals of Operations Research, 201 (2012), 449-473.  doi: 10.1007/s10479-012-1211-5.

[28]

T. K. Siu, A BSDE approach to optimal investment of an insurer with hidden regime switching, Stochastic Analysis and Applications, 31 (2013), 1-18.  doi: 10.1080/07362994.2012.727144.

[29]

T. K. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, 2015 (2015), Article ID 462524, 11pp. doi: 10.1155/2015/462524.

[30]

L. R. Sotomayor and A. Cadenillas, Explicit solution of consumption-investment problems in financial markets with regime switching, Mathematical Finance, 9 (2009), 251-279.  doi: 10.1111/j.1467-9965.2009.00366.x.

[31]

H. Tong, On a threshold model, in Pattern Recognition and Signal Processing (ed. C. H. Chen), NATO ASI Series E: Applied Sc., No. 29, Sijthoff & Noordhoff, The Netherlands, 1978,575-586. doi: 10.1007/978-94-009-9941-1_24.

[32] H. Tong, Threshold Models in Non-linear Time Series Analysis, Springer-Verlag, Berlin, 1978.  doi: 10.1007/978-1-4684-7888-4.
[33]

G. Yin and X. Zhou, Markowitz's mean-variance portfolio selection with regime switching: From discrete-time models to their continuous-time limits, IEEE Transactions Automatic Control, 49 (2004), 349-360.  doi: 10.1109/TAC.2004.824479.

[34]

K. F. C. YiuJ. LiuT. K. Siu and W. K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027.

[35]

X. ZhangT. K. Siu and Q. Meng, Porfolio selection in the enlarged Markovian regime-switching market, SIAM Journal on Control and Optimization, 48 (2010), 3368-3388.  doi: 10.1137/080736351.

[36]

X. ZhangR. J. Elliott and T. K. Siu, A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance, SIAM Journal on Control and Optimization, 50 (2012), 964-990.  doi: 10.1137/110839357.

[37]

X. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

show all references

References:
[1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd edition, Springer, New York, 2003. 
[2] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979.  doi: 10.1137/1.9780898719048.
[3]

J. M. C. Clark, The design of robust approximations to the stochastic differential equations for nonlinear filtering, in Communications Systems and Random Process Theory (ed. J. K. Skwirzynski), NATO Advanced Study Inst. Ser., Ser. E: Appl. Sci., No. 25, Sijthoff & Noordhoff, Alphen aan den Rijn, 1978,721-734.

[4] R. J. Elliott, Stochastic Calculus and Applications, Springer, Berlin-Heidelberg-New York, 1982. 
[5] R. J. ElliottL. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer, Berlin-Heidelberg-New York, 1994. 
[6]

R. J. ElliottT. K. Siu and A. Badescu, On mean-variance portfolio selection under a Markovian regime-switching model, Economic Modelling, 27 (2010), 678-686.  doi: 10.1016/j.econmod.2010.01.007.

[7]

R. J. Elliott and T. K. Siu, An HMM approach for optimal investment of an insurer, International Journal of Robust Nonlinear Control, 22 (2012), 778-807.  doi: 10.1002/rnc.1727.

[8] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin-Heidelberg-New York, 1975. 
[9]

S. M. Goldfeld and R. E. Quandt, A Markov model for switching regressions, Journal of Econometrics, 1 (1973), 3-16.  doi: 10.1016/0304-4076(73)90002-X.

[10]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.

[11]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384.  doi: 10.2307/1912559.

[12] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008. 
[13] M. JeanblancM. Yor and M. Chesney, Mathematical Methods for Financial Markets, Springer, New York, 2009.  doi: 10.1007/978-1-84628-737-4.
[14] G. Kallianpur, Stochastic Filtering Theory, Springer, Berlin-Heidelberg-New York, 1980. 
[15]

R. KornT. K. Siu and A. Zhang, Asset allocation for a DC pension fund under regime-switching environment, European Actuarial Journal, 1 (2011), 361-377.  doi: 10.1007/s13385-011-0021-5.

[16]

H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.

[17]

R. C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time model, The Review of Economics and Statistics, 51 (1969), 247-257. 

[18]

R. C. Merton, Optimal consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.

[19]

R. C. Merton, On estimating the expected return on the market, Journal of Financial Economics, 8 (1980), 323-361.  doi: 10.1016/0304-405X(80)90007-0.

[20]

W. Putschögl and J. Sass, Optimal consumption and investment under partial information, Decisions in Economics and Finance, 31 (2008), 137-170.  doi: 10.1007/s10203-008-0082-3.

[21]

R. E. Quandt, The estimation of parameters of linear regression system obeying two separate regimes, Journal of the American Statistical Association, 55 (1958), 873-880.  doi: 10.1080/01621459.1958.10501484.

[22]

U. Rieder and N. Bäuerle, Portfolio optimization with unobservable Markov-modulated drift process, Journal of Applied Probability, 42 (2005), 362-378.  doi: 10.1239/jap/1118777176.

[23]

J. Sass and U.G. Haussmann, Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain, Finance and Stochastics, 8 (2004), 553-577.  doi: 10.1007/s00780-004-0132-9.

[24]

A. SchiedH. Föllmer and S. Weber, Robust preferences and robust portfolio choice, Handbook of Numerical Analysis, 15 (2009), 29-87.  doi: 10.1016/S1570-8659(08)00002-1.

[25]

Y. Shen and T. K. Siu, Asset allocation under stochastic interest rate with regime switching, Economic Modelling, 29 (2012), 1126-1136.  doi: 10.1016/j.econmod.2012.03.024.

[26]

T. K. Siu, Long-term strategic asset allocation with inflation risk and regime switching, Quantitative Finance, 11 (2011), 1565-1580.  doi: 10.1080/14697680903055588.

[27]

T. K. Siu, A BSDE approach to risk-based asset allocation of pension funds with regime switching, Annals of Operations Research, 201 (2012), 449-473.  doi: 10.1007/s10479-012-1211-5.

[28]

T. K. Siu, A BSDE approach to optimal investment of an insurer with hidden regime switching, Stochastic Analysis and Applications, 31 (2013), 1-18.  doi: 10.1080/07362994.2012.727144.

[29]

T. K. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, 2015 (2015), Article ID 462524, 11pp. doi: 10.1155/2015/462524.

[30]

L. R. Sotomayor and A. Cadenillas, Explicit solution of consumption-investment problems in financial markets with regime switching, Mathematical Finance, 9 (2009), 251-279.  doi: 10.1111/j.1467-9965.2009.00366.x.

[31]

H. Tong, On a threshold model, in Pattern Recognition and Signal Processing (ed. C. H. Chen), NATO ASI Series E: Applied Sc., No. 29, Sijthoff & Noordhoff, The Netherlands, 1978,575-586. doi: 10.1007/978-94-009-9941-1_24.

[32] H. Tong, Threshold Models in Non-linear Time Series Analysis, Springer-Verlag, Berlin, 1978.  doi: 10.1007/978-1-4684-7888-4.
[33]

G. Yin and X. Zhou, Markowitz's mean-variance portfolio selection with regime switching: From discrete-time models to their continuous-time limits, IEEE Transactions Automatic Control, 49 (2004), 349-360.  doi: 10.1109/TAC.2004.824479.

[34]

K. F. C. YiuJ. LiuT. K. Siu and W. K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027.

[35]

X. ZhangT. K. Siu and Q. Meng, Porfolio selection in the enlarged Markovian regime-switching market, SIAM Journal on Control and Optimization, 48 (2010), 3368-3388.  doi: 10.1137/080736351.

[36]

X. ZhangR. J. Elliott and T. K. Siu, A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance, SIAM Journal on Control and Optimization, 50 (2012), 964-990.  doi: 10.1137/110839357.

[37]

X. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

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