American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A DC programming approach for sensor network localization with uncertainties in anchor positions
  • JIMO Home
  • This Issue
  • Next Article
    Solving structural engineering design optimization problems using an artificial bee colony algorithm
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

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 & 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.   Google Scholar

[2]

S. Browne, Stochastic differential portfolio games/em>,, Journal of Applied Probability, 37 (2000), 126.  doi: 10.1239/jap/1014842273.  Google Scholar

[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.   Google Scholar

[4]

R. Elliott, The existence of value in stochastic differential games,, SIAM: Journal of Control and Optimizaiton, 14 (1976), 85.  doi: 10.1137/0314006.  Google Scholar

[5]

R. Elliott and J. Hoek, An application of hidden Markov models to asset allocation problems,, Finance and Stochastics, 1 (1997), 229.  doi: 10.1007/s007800050022.  Google Scholar

[6]

R. Elliott and P. Kopp, Mathematics of Financial Markets,, $2^{nd}$ edition, (2005).   Google Scholar

[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.  doi: 10.1080/13504860600659222.  Google Scholar

[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.  doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[9]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Second edition. Stochastic Modelling and Applied Probability, (2006).   Google Scholar

[10]

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

[11]

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

[12]

C. Hipp and M. Plum, Optimal investment for insurers,, Insurance: Mathematics and Economics, 27 (2000), 215.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[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.  doi: 10.1016/0022-247X(69)90199-1.  Google Scholar

[14]

H. J. Kusher, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations,, Academic Press, (1977).   Google Scholar

[15]

H. J. Kusher, Approximations and Weak Convergence Methods for Random Processes,, MIT Press, (1984).   Google Scholar

[16]

H. J. Kusher, Numerical methods for stochastic control problems in continuous time,, SIAM: Journal of Control and Optimization, 28 (1990), 990.  doi: 10.1137/0328056.  Google Scholar

[17]

H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, Springer-Verlag: New York, (2001).   Google Scholar

[18]

H. J. Kushner and S. G. Chamberlain, Finite state stochastic games: existence theorems and computational procedures,, IEEE Trans. Automat. Control, 14 (1969), 248.   Google Scholar

[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.  doi: 10.3934/jimo.2013.9.487.  Google Scholar

[20]

S. Pliska, Introduction to Mathematical Finance,, United States: Blackwell Publishing, (1997).   Google Scholar

[21]

P. E. Protter, Stochastic Integration and Differential Equations,, 2nd edition, (2005).   Google Scholar

[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.  doi: 10.1109/TAC.2007.915169.  Google Scholar

[23]

S. Wan, Stochastic differential portfolio games based on utility with regime switching model,, in, (2007), 2302.   Google Scholar

[24]

H. Yang and L.Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance: Mathematics and Economics, 37 (2005), 615.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[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.  doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.673.  Google Scholar

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.   Google Scholar

[2]

S. Browne, Stochastic differential portfolio games/em>,, Journal of Applied Probability, 37 (2000), 126.  doi: 10.1239/jap/1014842273.  Google Scholar

[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.   Google Scholar

[4]

R. Elliott, The existence of value in stochastic differential games,, SIAM: Journal of Control and Optimizaiton, 14 (1976), 85.  doi: 10.1137/0314006.  Google Scholar

[5]

R. Elliott and J. Hoek, An application of hidden Markov models to asset allocation problems,, Finance and Stochastics, 1 (1997), 229.  doi: 10.1007/s007800050022.  Google Scholar

[6]

R. Elliott and P. Kopp, Mathematics of Financial Markets,, $2^{nd}$ edition, (2005).   Google Scholar

[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.  doi: 10.1080/13504860600659222.  Google Scholar

[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.  doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[9]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Second edition. Stochastic Modelling and Applied Probability, (2006).   Google Scholar

[10]

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

[11]

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

[12]

C. Hipp and M. Plum, Optimal investment for insurers,, Insurance: Mathematics and Economics, 27 (2000), 215.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[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.  doi: 10.1016/0022-247X(69)90199-1.  Google Scholar

[14]

H. J. Kusher, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations,, Academic Press, (1977).   Google Scholar

[15]

H. J. Kusher, Approximations and Weak Convergence Methods for Random Processes,, MIT Press, (1984).   Google Scholar

[16]

H. J. Kusher, Numerical methods for stochastic control problems in continuous time,, SIAM: Journal of Control and Optimization, 28 (1990), 990.  doi: 10.1137/0328056.  Google Scholar

[17]

H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, Springer-Verlag: New York, (2001).   Google Scholar

[18]

H. J. Kushner and S. G. Chamberlain, Finite state stochastic games: existence theorems and computational procedures,, IEEE Trans. Automat. Control, 14 (1969), 248.   Google Scholar

[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.  doi: 10.3934/jimo.2013.9.487.  Google Scholar

[20]

S. Pliska, Introduction to Mathematical Finance,, United States: Blackwell Publishing, (1997).   Google Scholar

[21]

P. E. Protter, Stochastic Integration and Differential Equations,, 2nd edition, (2005).   Google Scholar

[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.  doi: 10.1109/TAC.2007.915169.  Google Scholar

[23]

S. Wan, Stochastic differential portfolio games based on utility with regime switching model,, in, (2007), 2302.   Google Scholar

[24]

H. Yang and L.Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance: Mathematics and Economics, 37 (2005), 615.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[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.  doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.673.  Google Scholar

[1]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[2]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[3]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[4]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[5]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[6]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[7]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[8]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200
[9]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[10]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[11]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[12]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006
[13]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637
[14]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
[15]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29
[16]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[17]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181
[18]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[19]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[20]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences