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**Continuous-time mean-variance portfolio selection with liability and regime switching.**
*(English)*
Zbl 1231.91417

Summary: A continuous-time mean-variance model for individual investors with stochastic liability in a Markovian regime switching financial market, is investigated as a generalization of the model of X.Y. Zhou and G. Yin [SIAM J. Control Optim. 42, 1466–1482 (2003; Zbl 1175.91169)]. We assume that the risky stock’s price is governed by a Markovian regime-switching geometric Brownian motion, and the liability follows a Markovian regime-switching Brownian motion with drift, respectively. The evolution of appreciation rates, volatility rates and the interest rates are modulated by the Markov chain, and the Markov switching diffusion is assumed to be independent of the underlying Brownian motion. The correlation between the risky asset and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. Using the Lagrange multiplier technique and the linear-quadratic control technique, we get the expressions of the optimal portfolio and the mean-variance efficient frontier in closed forms. Further, the results of our special case without liability is consistent with those results of Zhou and Yin [loc. cit.].

### MSC:

91G10 | Portfolio theory |

91G80 | Financial applications of other theories |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

60J60 | Diffusion processes |

### Keywords:

continuous-time; mean; variance model; asset-liability management; Markov chain; linear-quadratic control### Citations:

Zbl 1175.91169
Full Text:
DOI

### References:

[1] | Bjork, T., Finite dimensional optimal filters for a class of \(It^o\)-processes with jumping parameters, Stochastics, 4, 167-183 (1980) · Zbl 0443.60038 |

[2] | Boyle, P.; Draviam, T., Pricing exotic options under regime switching, Insurance: Mathematics and Economics, 40, 267-282 (2007) · Zbl 1141.91420 |

[3] | Cajueiro, D. O.; Yoneyama, T., Optimal portfolio and consumption in a switching diffusion market, Brazilian Review of Econometrics, 24, 2, 227-248 (2004) |

[4] | Cheung, K. C.; Yang, H. L., Optimal investment-consumption strategy in a discrete-time model with regime switching, Discrete and Continuous Dynamical Systems. Series B, 8, 2, 315-332 (2007) · Zbl 1151.91491 |

[5] | Chiu, M. C.; Li, D., Asset and liability management under a continuous-time mean-variance optimization framework, Insurance: Mathematics and Economics, 39, 3, 330-355 (2006) · Zbl 1151.91493 |

[7] | Consiglio, A.; Cocco, F.; Zenios, S. A., Asset and liability modelling for participating policies with guarantees, European Journal of Operational Research, 186, 380-404 (2008) · Zbl 1138.91491 |

[8] | Decamps, M.; Schepper, A. D.; Goovaerts, M., A path integral approach to asset-liability management, Physica A: Statistical Mechanics and its Applications, 363, 2, 404-416 (2006) |

[9] | Fleming, W. H.; Soner, H. M., Controlled Markov Processes and Viscosity Solutions (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0773.60070 |

[10] | Hamilton, J. D., A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, 2, 357-384 (1989) · Zbl 0685.62092 |

[11] | Gerber, H. U.; Shiu, E. S.W., Geometric Brownian motion models for assets and liabilities: From pension funding to optimal dividends, North American Actuarial Journal, 7, 3, 37-56 (2004) · Zbl 1084.91517 |

[12] | Henderson, V.; Hobson, D., Utility Indifference Pricing: An Overview (2004), Princeton University Press: Princeton University Press New Jersey · Zbl 1158.91379 |

[13] | Josa-Fombellida, R.; Rincon-Zapatero, J. P., Mean-variance portfolio and contribution selection in stochastic pension funding, European Journal of Operational Research, 187, 120-137 (2008) · Zbl 1135.91019 |

[14] | Koo, H. K., consumption and portfolio selection with labor income: A continuous time approach, Mathematical Finance, 8, 1, 49-65 (1998) · Zbl 0911.90030 |

[15] | Leippold, M.; Trojani, F.; Vanini, P., A geometric approach to multi-period mean-variance optimization of assets and liabilities, Journal of Economics Dynamics and Control, 28, 1079-1113 (2004) · Zbl 1179.91234 |

[16] | Li, D.; Ng, W. L., Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10, 3, 387-406 (2000) · Zbl 0997.91027 |

[17] | Lim, A. E.B.; Zhou, X. Y., Mean-variance portfolio selection with random coefficients in a complete market, Mathematical and Operation Research, 27, 101-120 (2002) · Zbl 1082.91521 |

[18] | Liptser, R. S.; Shiryayev, A. N., Statistics of Random Process: General Theory, vol. 1 (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0364.60004 |

[19] | Luenberger, D. G., Optimization by Vector Space Methods (1968), John Wiley: John Wiley New York · Zbl 0184.44502 |

[20] | Markowitz, H., Portfolio selection, Journal of Finance, 7, 77-91 (1952) |

[21] | Merton, R. C., Lifetime portfolio selection under uncertainty: The continuous case, Review of Economics and Statistics, 51, 247-257 (1969) |

[22] | Naik, V., Option valuation and hedging strategies with jumps in the volatility of asset returns, Journal of Economic Theory, 48, 5, 1969-1984 (1993), 1993 |

[24] | Norberg, R., Ruin problems with assets and liabilities of diffusion type, Stochastic Processes and their Application, 81, 255-269 (1999) · Zbl 0962.60075 |

[25] | Siu, T. K., A game theoretic approach to option valuation under Markovian regime-switching models, Insurance: Mathematics and Economics, 42, 1146-1158 (2008) · Zbl 1141.91344 |

[26] | Xie, S. X.; Li, Z. F.; Wang, S. Y., Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach, Insurance: Mathematics and Economics, 42, 943-953 (2008) · Zbl 1141.91474 |

[28] | Yong, J.; Zhou, X. Y., Stochastic Controls: Hamiltonian Systems and HJB Equations (1999), Springer-Verlag · Zbl 0943.93002 |

[29] | Zhang, Q., Stock trading: An optimal selling rule, SIAM Journal on Control and Optimization, 40, 1, 64-87 (2001) · Zbl 0990.91014 |

[30] | Zhou, X. Y.; Li, D., Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42, 19-33 (2000) · Zbl 0998.91023 |

[31] | Zhou, X. Y.; Yin, G., Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42, 4, 1466-1482 (2003) · Zbl 1175.91169 |

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