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OPTIMAL PORTFOLIO AND CONSUMPTION FOR A MARKOVIAN REGIME-SWITCHING JUMP-DIFFUSION PROCESS

Published online by Cambridge University Press:  21 July 2021

CAIBIN ZHANG*
Affiliation:
School of Finance, Nanjing University of Finance and Economics, Nanjing210023, China
ZHIBIN LIANG
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing210023, China; e-mail: liangzhibin111@hotmail.com.
KAM CHUEN YUEN
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China; e-mail: kcyuen@hku.hk.

Abstract

We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases.

Type
Research Article
Copyright
© Australian Mathematical Society 2021

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References

Aase, K., “Optimal portfolio diversification in a general continuous-time model”, Stochastic Process. Appl. 18 (1984) 8198; doi:10.1016/0304-4149(84)90163-7.CrossRefGoogle Scholar
Akian, M., Menaldi, J. and Sulem, A., “On an investment–consumption model with transaction costs”, SIAM J. Control Optim. 34 (1996) 329364; doi:10.1137/S0363012993247159.CrossRefGoogle Scholar
Cajueiro, D. O. and Yoneyama, T., “Optimal portfolio and consumption in a switching diffusion market”, Braz. Rev. Econometrics 24 (2004) 227247; doi:10.12660/bre.v24n22004.2711.CrossRefGoogle Scholar
Chacko, G. and Viceira, L., “Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets”, Rev. Financ. Stud. 18 (2005) 13691402; doi:10.1093/rfs/hhi035.CrossRefGoogle Scholar
Cheridito, P. and Hu, Y., “Optimal consumption and investment in incomplete markets with general constraints”, Stoch. Dyn. 11 (2011) 283299; doi:10.1142/S0219493711003280.CrossRefGoogle Scholar
Collin-Dufresne, P., Daniel, K. D. and Sağlam, M., “Liquidity regimes and optimal dynamic asset allocation”, J. Financ. Econ. 136 (2020) 379406; doi:10.1016/j.jfineco.2019.09.011.CrossRefGoogle Scholar
Cont, R. and Tankov, P., Financial modelling with jump processes (CRC Press, Boca Raton, FL, 2004), available at https://3lib.net/book/715906/e64752.Google Scholar
Fleming, W. H. and Soner, H. M., Controlled Markov processes and viscosity solutions, 2nd edn (Springer, New York, 2006), available at https://3lib.net/book/494957/a3ce5b.Google Scholar
Framstad, N. C., Øksendal, B. and Sulem, A., “Optimal consumption and portfolio in a jump diffusion market”, INRIA, Paris, 1998, 9–20, working paper, available at http://hdl.handle.net/11250/163833.Google Scholar
Framstad, N. C., Øksendal, B. and Sulem, A., “Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs”, J. Math. Econom. 35 (2001) 233257; doi:10.1016/S0304-4068(00)00067-7.CrossRefGoogle Scholar
Framstad, N. C., Øksendal, B. and Sulem, A., “Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance”, J. Optim. Theory Appl. 121 (2004) 7798; doi:10.1023/B:JOTA.0000026132.62934.96.CrossRefGoogle Scholar
Gassiat, P., Gozzi, F. and Pham, H., “Investment/consumption problem in illiquid markets with regime switching”, SIAM J. Control Optim. 52 (2014) 17611786; doi:10.1137/120876976.CrossRefGoogle Scholar
Gille, S. and Hornbostel, J., “A zero theorem for the transfer of coherent Witt groups”, Math. Nachr. 278 (2005) 815823; doi:10.1002/mana.200310274.CrossRefGoogle Scholar
Guambe, C. and Kufakunesu, R., “A note on optimal investment–consumption–insurance in a Lévy market”, Insurance Math. Econom. 65 (2015) 3036; doi:10.1016/j.insmatheco.2015.07.008.CrossRefGoogle Scholar
Hu, F. and Wang, R., “Optimal investment–consumption strategy with liability and regime switching model under value-at-risk constraint”, Appl. Math. Comput. 313 (2017) 103108; doi:10.1016/j.amc.2017.04.034.Google Scholar
Jonathan, E. and Ingersoll, J., “Optimal consumption and portfolio rules with intertemporally dependent utility of consumption”, J. Econom. Dynam. Control 16 (1992) 781–712; doi:10.1016/0165-1889(92)90054-I.Google Scholar
Karatzas, I. and Žitković, G., “Optimal consumption from investment and random endowment in incomplete semimartingale markets”, Ann. Probab . 31 (2003) 18211858, available at https://projecteuclid.org/euclid.aop/1068646367.CrossRefGoogle Scholar
Koo, H., “Consumption and portfolio selection with labor income: a continuous time approach”, Math. Finance 8 (1998) 4965; doi:10.1111/1467-9965.00044.CrossRefGoogle Scholar
Kronborg, M. T. and Steffensen, M., “Optimal consumption, investment and life insurance with surrender option guarantee”, Scand. Actuar. J. 1 (2015) 5987; doi:10.1080/03461238.2013.775964.CrossRefGoogle Scholar
Liu, R., “Optimal investment and consumption with proportional transaction costs in regime-switching model”, J. Optim. Theory Appl. 163 (2014) 614641; doi:10.1007/s10957-013-0445-y.CrossRefGoogle Scholar
Ma, G., Siu, C. C. and Zhu, S. P., “Optimal investment and consumption with return predictability and execution costs”, Econom. Model. 88 (2020) 408419; doi:10.1016/j.econmod.2019.09.051.CrossRefGoogle Scholar
Merton, R. C., “Lifetime portfolio selection under uncertainty: the continuous-time case”, Rev. Econom. Statist. 51 (1969) 247257; doi:10.2307/1926560.CrossRefGoogle Scholar
Merton, R. C., “Optimal consumption and portfolio rules in a continuous time model”, J. Econom. Theory 3 (1971) 373413; doi:10.1016/0022-0531(71)90038-X.CrossRefGoogle Scholar
Merton, R. C., “Option pricing when underlying stock returns are discontinuous”, J. Financ. Econ. 3 (1976) 125144; doi:10.1016/0304-405X(76)90022-2.CrossRefGoogle Scholar
Nguyen, T., “Optimal investment and consumption with downside risk constraint in jump-diffusion models”, Preprint, 2016, arXiv:1604.05584.Google Scholar
Perera, R., “Optimal investment, consumption-leisure, insurance and retirement choice”, Ann. Finance 9 (2013) 689723; doi:10.1007/s10436-012-0214-1.CrossRefGoogle Scholar
Ruan, X., Zhu, W. and Huang, J., “Optimal portfolio and consumption with habit formation in a jump diffusion market”, Appl. Math. Comput. 222 (2013) 391401; doi:10.1016/j.amc.2013.07.063.Google Scholar
Schied, A., “Robust optimal control for a consumption–investment problem”, Math. Methods Oper. Res. 67 (2008) 120; doi:10.1007/s00186-007-0172-y.CrossRefGoogle Scholar
Shen, Y. and Wei, J., “Optimal investment–consumption–insurance with random parameters”, Scand. Actuar. J. 1 (2016) 3762; doi:10.1080/03461238.2014.900518.CrossRefGoogle Scholar
Sotomayor, L. R. and Cadenillas, A., “Explicit solutions of consumption–investment problems in financial markets with regime switching”, Math. Finance 19 (2009) 251279; doi:10.1111/j.1467-9965.2009.00366.x.CrossRefGoogle Scholar
Wachter, J., “Portfolio and consumption decisions under mean-reverting returns: an exact solution for complete markets”, J. Financ. Quant. Anal. 37 (2002) 7391; doi:10.2307/3594995.CrossRefGoogle Scholar
Yong, J. and Zhou, X., Stochastic controls: Hamiltonian systems and HJB equations (Springer, New York, 1999), available at https://3lib.net/book/1184248/5c0026.CrossRefGoogle Scholar
Zhang, X., Elliott, R. J. and Siu, T. K., “A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance”, SIAM J. Control Optim. 50 (2012) 964990; doi:10.1137/110839357.CrossRefGoogle Scholar