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**An investment and consumption problem with CIR interest rate and stochastic volatility.**
*(English)*
Zbl 1291.91189

Summary: We are concerned with an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is described by the Cox-Ingersoll-Ross (CIR) model and the volatility of the stock is driven by Heston’s stochastic volatility model. We apply stochastic optimal control theory to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function and choose power utility and logarithm utility for our analysis. By using separate variable approach and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategy. A numerical example is given to illustrate our results and to analyze the effect of market parameters on the optimal investment and consumption strategies.

### MSC:

91G10 | Portfolio theory |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

91B70 | Stochastic models in economics |

93E20 | Optimal stochastic control |

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\textit{H. Chang} and \textit{X.-M. Rong}, Abstr. Appl. Anal. 2013, Article ID 219397, 12 p. (2013; Zbl 1291.91189)

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### References:

[1] | Merton, R. C., Lifetime portfolio selection under uncertainty: the continuous-time case, The Review of Economics and Statistics, 51, 3, 247-257 (1969) |

[2] | Merton, R. C., Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3, 4, 373-413 (1971) · Zbl 1011.91502 |

[3] | Fleming, W. H.; Zariphopoulou, T., An optimal investment/consumption model with borrowing, Mathematics of Operations Research, 16, 4, 802-822 (1991) · Zbl 0744.90004 |

[4] | Vila, J.-L.; Zariphopoulou, T., Optimal consumption and portfolio choice with borrowing constraints, Journal of Economic Theory, 77, 2, 402-431 (1997) · Zbl 0897.90078 |

[5] | Duffie, D.; Fleming, W.; Soner, H. M.; Zariphopoulou, T., Hedging in incomplete markets with HARA utility, Journal of Economic Dynamics & Control, 21, 4-5, 753-782 (1997) · Zbl 0899.90026 |

[6] | Dumas, B.; Luciano, E., An exact solution to a dynamic portfolio choice problem under transactions costs, The Journal of Finance, 46, 2, 577-595 (1991) |

[7] | Shreve, S. E.; Soner, H. M., Optimal investment and consumption with transaction costs, The Annals of Applied Probability, 4, 3, 609-692 (1994) · Zbl 0813.60051 |

[8] | Liu, H.; Loewenstein, M., Optimal portfolio selection with transaction costs and finite horizons, Review of Financial Studies, 15, 3, 805-835 (2002) |

[9] | Dai, M.; Jiang, L.; Li, P.; Yi, F., Finite horizon optimal investment and consumption with transaction costs, SIAM Journal on Control and Optimization, 48, 2, 1134-1154 (2009) · Zbl 1189.35166 |

[10] | Ho, T. S. Y.; Lee, S. B., Term structure movements and pricing interest contingent claims, Journal of Finance, 41, 5, 1011-1029 (1986) |

[11] | Vasicek, O., An equilibrium characterization of the term structure, Journal of Financial Economics, 5, 2, 177-188 (1977) · Zbl 1372.91113 |

[12] | Cox, J. C.; Ingersoll,, J. E.; Ross, S. A., A theory of the term structure of interest rates, Econometrica. Journal of the Econometric Society, 53, 2, 385-407 (1985) · Zbl 1274.91447 |

[13] | Korn, R.; Kraft, H., A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40, 4, 1250-1269 (2001/02) · Zbl 1020.93029 |

[14] | Grasselli, M., A stability result for the HARA class with stochastic interest rates, Insurance: Mathematics & Economics, 33, 3, 611-627 (2003) · Zbl 1103.91349 |

[15] | Deelstra, G.; Grasselli, M.; Koehl, P.-F., Optimal investment strategies in the presence of a minimum guarantee, Insurance: Mathematics & Economics, 33, 1, 189-207 (2003) · Zbl 1074.91013 |

[16] | Josa-Fombellida, R.; Rincón-Zapatero, J. P., Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates, European Journal of Operational Research, 201, 1, 211-221 (2010) · Zbl 1177.91125 |

[17] | Gao, J., Stochastic optimal control of DC pension funds, Insurance: Mathematics & Economics, 42, 3, 1159-1164 (2008) · Zbl 1141.91439 |

[18] | Deelstra, G.; Grasselli, M.; Koehl, P.-F., Optimal investment strategies in a CIR framework, Journal of Applied Probability, 37, 4, 936-946 (2000) · Zbl 0989.91040 |

[19] | Fleming, W. H.; Pang, T., An application of stochastic control theory to financial economics, SIAM Journal on Control and Optimization, 43, 2, 502-531 (2004) · Zbl 1101.93085 |

[20] | Munk, C.; Sørensen, C., Optimal consumption and investment strategies with stochastic interest rates, Journal of Banking and Finance, 28, 8, 1987-2013 (2004) |

[21] | Heston, S. L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6, 2, 327-343 (1993) · Zbl 1384.35131 |

[22] | Kraft, H., Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility, Quantitative Finance, 5, 3, 303-313 (2005) · Zbl 1134.91438 |

[23] | Fleming, W. H.; Hernández-Hernández, D., An optimal consumption model with stochastic volatility, Finance and Stochastics, 7, 2, 245-262 (2003) · Zbl 1035.60028 |

[24] | Chacko, G.; Viceira, L. M., Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, Review of Financial Studies, 18, 4, 1369-1402 (2005) |

[25] | Li, Z. F.; Zeng, Y.; Lai, Y. Z., Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model, Insurance, Mathematics and Economics, 50, 1, 191-203 (2012) · Zbl 1284.91250 |

[26] | Sasha, F. S.; Zariphopoulou, T., Dynamic asset allocation and consumption choice in incomplete markets, Australian Economic Papers, 44, 4, 414-454 (2005) |

[27] | Castañeda-Leyva, N.; Hernández-Hernández, D., Optimal consumption-investment problems in incomplete markets with stochastic coefficients, SIAM Journal on Control and Optimization, 44, 4, 1322-1344 (2005) · Zbl 1140.91381 |

[28] | Liu, J., Portfolio selection in stochastic environments, Review of Financial Studies, 20, 1, 1-39 (2007) |

[29] | Michelbrink, D.; Le, H., A martingale approach to optimal portfolios with jump-diffusions, SIAM Journal on Control and Optimization, 50, 1, 583-599 (2012) · Zbl 1251.91055 |

[30] | Li, J.; Wu, R., Optimal investment problem with stochastic interest rate and stochastic volatility: maximizing a power utility, Applied Stochastic Models in Business and Industry, 25, 3, 407-420 (2009) · Zbl 1224.91140 |

[31] | Noh, E.-J.; Kim, J.-H., An optimal portfolio model with stochastic volatility and stochastic interest rate, Journal of Mathematical Analysis and Applications, 375, 2, 510-522 (2011) · Zbl 1202.91302 |

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