Optimal investment and consumption for an insurer with high-watermark performance fee.

*(English)*Zbl 1394.91237Summary: The optimal investment and consumption problem is investigated for an insurance company, which is subject to the payment of high-watermark fee from profit. The objective of insurance company is to maximize the expected cumulated discount utility up to ruin time. The consumption behavior considered in this paper can be viewed as dividend payment of the insurance company. It turns out that the value function of the proposed problem is the viscosity solution to the associated HJB equation. The regularity of the viscosity is discussed and some asymptotic results are provided. With the help of the smooth properties of viscosity solutions, we complete the verification theorem of the optimal control policies and the potential applications of the main result are discussed.

PDF
BibTeX
XML
Cite

\textit{L. Xu} et al., Math. Probl. Eng. 2015, Article ID 413072, 14 p. (2015; Zbl 1394.91237)

Full Text:
DOI

##### References:

[1] | 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 |

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

[3] | 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 |

[4] | Zariphopoulou, T., Optimal investment and consumption models with non-linear stock dynamics, Mathematical Methods of Operations Research, 50, 2, 271-296, (1999) · Zbl 0961.91016 |

[5] | Øksendal, B.; Sulem, A., Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM Journal on Control and Optimization, 40, 6, 1765-1790, (2002) · Zbl 1102.91054 |

[6] | Fleming, W. H.; Pang, T., A stochastic control model of investment, production and consumption, Quarterly of Applied Mathematics, 63, 1, 71-87, (2005) · Zbl 1080.91031 |

[7] | Karatzas, I.; Shreve, S. E., Methods of Mathematical Finance, 39, (1998), Springer · Zbl 0941.91032 |

[8] | Davis, M. H.; Norman, A. R., Portfolio selection with transaction costs, Mathematics of Operations Research, 15, 4, 676-713, (1990) · Zbl 0717.90007 |

[9] | Janeček, K.; Sîrbu, M., Optimal investment with high-watermark performance fee, SIAM Journal on Control and Optimization, 50, 2, 790-819, (2012) · Zbl 1248.91092 |

[10] | Whalley, A. E.; Wilmott, P., An asymptotic analysis of an optimal hedging model for option pricing with transaction costs, Mathematical Finance, 7, 3, 307-324, (1997) · Zbl 0885.90019 |

[11] | Cvitanić, J.; Karatzas, I., Hedging and portfolio optimization under transaction costs: a martingale approach, Mathematical Finance, 6, 2, 133-165, (1996) · Zbl 0919.90007 |

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

[13] | Korn, R., Portfolio optimisation with strictly positive transaction costs and impulse control, Finance and Stochastics, 2, 2, 85-114, (1998) · Zbl 0894.90021 |

[14] | Obizhaeva, A. A.; Wang, J., Optimal trading strategy and supply/demand dynamics, Journal of Financial Markets, 16, 1, 1-32, (2013) |

[15] | Stiglitz, J. E., Some aspects of the taxation of capital gains, Journal of Public Economics, 21, 2, 257-294, (1983) |

[16] | Dammon, R. M.; Spatt, C. S.; Zhang, H. H., Optimal consumption and investment with capital gains taxes, Review of Financial Studies, 14, 3, 583-616, (2001) |

[17] | Goetzmann, W. N.; Ingersoll, J. E.; Ross, S. A., High-water marks and hedge fund management contracts, The Journal of Finance, 58, 4, 1685-1718, (2003) |

[18] | Guasoni, P.; Wang, G., High-water marks and separation of private investments, SSRN Electronic Journal, (2012) |

[19] | Højgaard, B.; Taksar, M., Optimal proportional reinsurance policies for diffusion models with transaction costs, Insurance: Mathematics and Economics, 22, 1, 41-51, (1998) · Zbl 1093.91518 |

[20] | Cairns, A., Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time, ASTIN Bulletin, 30, 1, 19-56, (2000) · Zbl 1018.91028 |

[21] | Zhu, J., Optimal dividend control for a generalized risk model with investment incomes and debit interest, Scandinavian Actuarial Journal, 2013, 2, 140-162, (2013) · Zbl 1308.91093 |

[22] | He, L.; Liang, Z., Optimal dynamic asset allocation strategy for ELA scheme of DC pension plan during the distribution phase, Insurance: Mathematics and Economics, 52, 2, 404-410, (2013) · Zbl 1284.91521 |

[23] | Bielecki, T. R.; Pliska, S. R., Risk sensitive asset management with transaction costs, Finance and Stochastics, 4, 1, 1-33, (2000) · Zbl 0982.91024 |

[24] | Dai, M.; Yi, F., Finite-horizon optimal investment with transaction costs: a parabolic double obstacle problem, Journal of Differential Equations, 246, 4, 1445-1469, (2009) · Zbl 1227.35182 |

[25] | Young, V. R., Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8, 4, 106-126, (2004) · Zbl 1085.60514 |

[26] | Grandell, J., Aspects of Risk Theory, (1991), Berlin, Germany: Springer, Berlin, Germany · Zbl 0717.62100 |

[27] | Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the cramér-lundberg model, Mathematical Finance, 15, 2, 261-308, (2005) · Zbl 1136.91016 |

[28] | Watanabe, S.; Ikeda, N., Stochastic Differential Equations and Diffusion Processes, (1981), Elsevier · Zbl 0495.60005 |

[29] | Crandall, M. G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27, 1, 1-67, (1992) · Zbl 0755.35015 |

[30] | Pham, H., Continuous-time Stochastic Control and Optimization with Financial Applications, 1, (2009), Springer · Zbl 1165.93039 |

[31] | Benth, F. E.; Karlsen, K. H.; Reikvam, K., Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs, Stochastics, 74, 3-4, 517-569, (2002) · Zbl 1035.91027 |

[32] | Barles, G.; Imbert, C., Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Annales de l’Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis, 25, 3, 567-585, (2008) · Zbl 1155.45004 |

[33] | Magill, M. J. P.; Constantinides, G. M., Portfolio selection with transactions costs, Journal of Economic Theory, 13, 2, 245-263, (1976) · Zbl 0361.90001 |

[34] | Dufresne, F.; Gerber, H. U., Risk theory for the compound poisson process that is perturbed by diffusion, Insurance Mathematics and Economics, 10, 1, 51-59, (1991) · Zbl 0723.62065 |

[35] | Yang, H.; Zhang, L., Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37, 3, 615-634, (2005) · Zbl 1129.91020 |

[36] | Protter, P. E., Stochastic Integration and Differential Equations: Version 2.1, 21, (2004), Springer |

[37] | Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20, 4, 937-958, (1995) · Zbl 0846.90012 |

[38] | Soner, H. M., Optimal control of jump-markov processes and viscosity solutions, Stochastic Differential Systems, Stochastic Control Theory and Applications, 501-511, (1988), Springer · Zbl 0850.93889 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.