Optimal stochastic investment games under Markov regime switching market

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July 2014, 10(3): 795-815. doi: 10.3934/jimo.2014.10.795

Optimal stochastic investment games under Markov regime switching market

Lin Xu ^1, , Rongming Wang ^2, and Dingjun Yao ^3,

1.	School of Mathematics and Computer Science, Anhui Normal University, Wuhu, Anhui, 241003, China
2.	School of Finance and Statistics, East China Normal University, Shanghai, 200241
3.	School of Finance, Nanjing University of Finance and Economics, Nanjing, Jiangsu, 210046, China

Received November 2012 Revised June 2013 Published November 2013

Abstract
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This paper focuses on stochastic investment games between two investors with incorporating the influence of the macro economical environment that modeled by a Markov chain with $d$ states. There are two correlated assets are available to two investors, each investor can only invest into one of assets and his opponent choose to invest the other one. The dynamic of the two assets are driven by two drifted Brownian motion with coefficients specified by the functions of the Markov chain. Thus the system considered in this paper is controlled SDEs with random coefficients. Only one payoff function is available to both investors, one investor wants to maximize the expected payoff function, while his opponent wants to minimize the quantity at the same time. As results, the existence of the saddle point of the game, a couple of equations satisfied by the value functions and optimal policies for both investors are derived. Based on finite-difference method and weak convergence theory, a vector-valued Markov chain is constructed for approximating the underlying risky process weakly, which enables us to obtain the value function and optimal policies numerically. To some extend, we can view this paper as a further research of the problems proposed in Wan [23].

Keywords: Optimal investment, finite difference method, Markov regime switching market, weak convergence., zero-sum stochastic differential games.

Mathematics Subject Classification: Primary: 91G10, 91G80; Secondary: 90B5.

Citation: Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial and Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795

References:

[1]	S. Browne, Optimal investment policy for a firm with random risk process: Exponential utility and minimizing the probabilty of ruin, Mathematical Operation Research, 20 (1995), 937-958.
[2]	S. Browne, Stochastic differential portfolio games/em>, Journal of Applied Probability, 37 (2000), 126-147. doi: 10.1239/jap/1014842273.
[3]	M. Clements and H. Krolzig, Can regime-swichting models reproducing the business cycle features of U.S. aggregate consumption, investment and output? International Journal of Financing and Economics, 9 (2004), 1-14.
[4]	R. Elliott, The existence of value in stochastic differential games, SIAM: Journal of Control and Optimizaiton, 14 (1976), 85-94. doi: 10.1137/0314006.
[5]	R. Elliott and J. Hoek, An application of hidden Markov models to asset allocation problems, Finance and Stochastics, 1 (1997), 229-238. doi: 10.1007/s007800050022.
[6]	R. Elliott and P. Kopp, Mathematics of Financial Markets, $2^{nd}$ edition, Springer-Verlag, New York, 2005.
[7]	R. Elliott, T. K. Siu and L. Chan, Pricing volatility swaps under Heston's stochastic volatility model with regime switching, Applied Mathematical Finance, 14 (2007), 41-62. doi: 10.1080/13504860600659222.
[8]	W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Universtiy Mathematical Journal, 38 (1989), 293-313. doi: 10.1512/iumj.1989.38.38015.
[9]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.
[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]	C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228. doi: 10.1016/S0167-6687(00)00049-4.
[13]	H. Kushner and S. Chamberlain, On stochastic differential games: Sufficient conditions that a given strategy be a saddle point, and numerical procedures for the solution of the game, Journal of Mathmatical Analysis and Applications, 26 (1969), 560-575. doi: 10.1016/0022-247X(69)90199-1.
[14]	H. J. Kusher, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.
[15]	H. J. Kusher, Approximations and Weak Convergence Methods for Random Processes, MIT Press, Cambirdge, 1984.
[16]	H. J. Kusher, Numerical methods for stochastic control problems in continuous time, SIAM: Journal of Control and Optimization, 28 (1990), 990-1048. doi: 10.1137/0328056.
[17]	H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer-Verlag: New York, 2nd edition, 2001.
[18]	H. J. Kushner and S. G. Chamberlain, Finite state stochastic games: existence theorems and computational procedures, IEEE Trans. Automat. Control, 14 (1969), 248-255.
[19]	H. Meng, F. Yuen, T. Siu and H. Yang, Optimal portfolio in a continuous-time self-exciting threshold model, Journal of Industrial and Management Optimization, 9 (2013), 487-504. doi: 10.3934/jimo.2013.9.487.
[20]	S. Pliska, Introduction to Mathematical Finance, United States: Blackwell Publishing, 1997.
[21]	P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springe-Verlag, 2005.
[22]	Q. Song, G. Yin and Z. Zhang, Numercial solutions for stochastic differential games with regime switching, IEEE Transactions on Automatica Control, 53 (2008), 509-521. doi: 10.1109/TAC.2007.915169.
[23]	S. Wan, Stochastic differential portfolio games based on utility with regime switching model, in "2007 IEEE International Conference on Control and Automation, Guangzhou, China", 2007, 2302-2305.
[24]	H. Yang and L.Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009.
[25]	D. Yao, H. Yang and R. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, Journal of Industrial and Management Optimization, 6 (2010), 761-777. doi: 10.3934/jimo.2010.6.761.
[26]	K. C. Yiu, J. Liu, T. Siu and 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.
[27]	Y. Zeng and Z. Li, Optimal reinsurance-investment strategies for insurers under mean-CaR criteria, Journal of Industrial and Management Optimization, 8 (2012), 673-690. doi: 10.3934/jimo.2012.8.673.

show all references

References:

[1]	S. Browne, Optimal investment policy for a firm with random risk process: Exponential utility and minimizing the probabilty of ruin, Mathematical Operation Research, 20 (1995), 937-958.
[2]	S. Browne, Stochastic differential portfolio games/em>, Journal of Applied Probability, 37 (2000), 126-147. doi: 10.1239/jap/1014842273.
[3]	M. Clements and H. Krolzig, Can regime-swichting models reproducing the business cycle features of U.S. aggregate consumption, investment and output? International Journal of Financing and Economics, 9 (2004), 1-14.
[4]	R. Elliott, The existence of value in stochastic differential games, SIAM: Journal of Control and Optimizaiton, 14 (1976), 85-94. doi: 10.1137/0314006.
[5]	R. Elliott and J. Hoek, An application of hidden Markov models to asset allocation problems, Finance and Stochastics, 1 (1997), 229-238. doi: 10.1007/s007800050022.
[6]	R. Elliott and P. Kopp, Mathematics of Financial Markets, $2^{nd}$ edition, Springer-Verlag, New York, 2005.
[7]	R. Elliott, T. K. Siu and L. Chan, Pricing volatility swaps under Heston's stochastic volatility model with regime switching, Applied Mathematical Finance, 14 (2007), 41-62. doi: 10.1080/13504860600659222.
[8]	W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Universtiy Mathematical Journal, 38 (1989), 293-313. doi: 10.1512/iumj.1989.38.38015.
[9]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.
[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]	C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228. doi: 10.1016/S0167-6687(00)00049-4.
[13]	H. Kushner and S. Chamberlain, On stochastic differential games: Sufficient conditions that a given strategy be a saddle point, and numerical procedures for the solution of the game, Journal of Mathmatical Analysis and Applications, 26 (1969), 560-575. doi: 10.1016/0022-247X(69)90199-1.
[14]	H. J. Kusher, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.
[15]	H. J. Kusher, Approximations and Weak Convergence Methods for Random Processes, MIT Press, Cambirdge, 1984.
[16]	H. J. Kusher, Numerical methods for stochastic control problems in continuous time, SIAM: Journal of Control and Optimization, 28 (1990), 990-1048. doi: 10.1137/0328056.
[17]	H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer-Verlag: New York, 2nd edition, 2001.
[18]	H. J. Kushner and S. G. Chamberlain, Finite state stochastic games: existence theorems and computational procedures, IEEE Trans. Automat. Control, 14 (1969), 248-255.
[19]	H. Meng, F. Yuen, T. Siu and H. Yang, Optimal portfolio in a continuous-time self-exciting threshold model, Journal of Industrial and Management Optimization, 9 (2013), 487-504. doi: 10.3934/jimo.2013.9.487.
[20]	S. Pliska, Introduction to Mathematical Finance, United States: Blackwell Publishing, 1997.
[21]	P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springe-Verlag, 2005.
[22]	Q. Song, G. Yin and Z. Zhang, Numercial solutions for stochastic differential games with regime switching, IEEE Transactions on Automatica Control, 53 (2008), 509-521. doi: 10.1109/TAC.2007.915169.
[23]	S. Wan, Stochastic differential portfolio games based on utility with regime switching model, in "2007 IEEE International Conference on Control and Automation, Guangzhou, China", 2007, 2302-2305.
[24]	H. Yang and L.Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009.
[25]	D. Yao, H. Yang and R. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, Journal of Industrial and Management Optimization, 6 (2010), 761-777. doi: 10.3934/jimo.2010.6.761.
[26]	K. C. Yiu, J. Liu, T. Siu and 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.
[27]	Y. Zeng and Z. Li, Optimal reinsurance-investment strategies for insurers under mean-CaR criteria, Journal of Industrial and Management Optimization, 8 (2012), 673-690. doi: 10.3934/jimo.2012.8.673.

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