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OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY

Part of: Mathematical finance

Published online by Cambridge University Press:  18 February 2019

L. LI  and
H. MI Open the ORCID record for H. MI [Opens in a new window]
L. LI
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu, China email perfectlyli@163.com, mihui@njnu.edu.cn
H. MI*
Affiliation:
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu, China email perfectlyli@163.com, mihui@njnu.edu.cn
*
✉mihui@njnu.edu.cn
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Abstract

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We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results.

Keywords

stochastic differential delay equationpower utility functionstochastic factorCox–Ingersoll–Ross model

MSC classification

Primary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Secondary: 91G10: Portfolio theory
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 1 , January 2019 , pp. 99 - 117
Copyright
© 2019 Australian Mathematical Society 

References

C.-X. Aand Li, Z.-F., “Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston’s SV model”, Insurance Math. Econom. 61 (2015) 181–196; doi:10.1016/j.insmatheco.2015.01.005.Google Scholar
C.-X. Aand Shao, Y., “Portfolio optimization problem with delay under Cox–Ingersoll–Ross model”, J. Math. Finance 7 (2017) 699–717; doi:10.4236/jmf.2017.73037.Google Scholar
Agram, N., Haadem, S., Øksendal, B. and Proske, F., “A maximum principle for infinite horizon delay equations”, SIAM J. Math. Anal. 45 (2013) 2499–2522; doi:10.1137/120882809.Google Scholar
Bielecki, T. R. and Pliska, S. R., “Risk-sensitive dynamic asset management”, Appl. Math. Optim. 39 (1999) 337–360; doi:10.1007/s002459900110.Google Scholar
Chacko, G. and Viceira, L. M., “Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets”, Rev. Financ. Stud. 18 (2005) 1369–1402; doi:10.1093/rfs/hhi035.Google Scholar
Chang, M.-H., Pang, T. and Yang, Y., “A stochastic portfolio optimization model with bounded memory”, Math. Oper. Res. 36 (2011) 604–619; doi:10.1287/moor.1110.0508.Google Scholar
Cox, J. C. and Huang, C. F., “Optimal consumption and portfolio policies when asset prices follow a diffusion process”, J. Econom. Theory 49 (1989) 33–83; doi:10.1016/0022-0531(89)90067-7.Google Scholar
Delong, Ł. and Klüppelberg, C., “Optimal investment and consumption in a Black–Scholes market with Lévy-driven stochastic coefficients”, Ann. Appl. Probab. 18 (2008) 879–908; doi:10.1214/07-AAP475.Google Scholar
Elsanosi, I., Øksendal, B. and Sulem, A., “Some solvable stochastic control problems with delay”, Stochastics 71 (2000) 69–89; doi:10.1080/17442500008834259.Google Scholar
Fleming, W. H. and Hernández-Hernández, D., “An optimal consumption model with stochastic volatility”, Finance Stoch. 7 (2003) 245–262; doi:10.1007/s0078002000.Google Scholar
Fouque, J.-P., Papanicolaou, G. and Sircar, K. R., Derivatives in financial markets with stochastic volatility (Cambridge University Press, Cambridge, 2000).Google Scholar
Hernández-Hernández, D. and Schied, A., “Robust utility maximization in a stochastic factor model”, Statist. Decisions 24 (2006) 109–125; doi:10.1524/stnd.2006.24.1.109.Google Scholar
Liu, J., “Portfolio selection in stochastic environments”, Rev. Financ. Stud. 20 (2007) 1–39; doi:10.1093/rfs/hhl001.Google Scholar
Liu, J. and Pan, J., “Dynamic derivative strategies”, J. Financ. Econ. 69 (2003) 401–430; doi:10.1016/S0304-405X(03)00118-1.Google Scholar
Merton, R. C., “Lifetime portfolio selection under uncertainty: the continuous-time case”, Rev. Econ. Stat. 51 (1969) 247–257; doi:10.2307/1926560.Google Scholar
Merton, R. C., “Optimum consumption and portfolio rules in a continuous-time model”, J. Econom. Theory 3 (1971) 373–413; doi:10.1016/0022-0531(71)90038-X.Google Scholar
Øksendal, B. and Sulem, A., “A maximum principle for optimal control of stochastic systems with delay, with applications to finance”, in: Optimal control and partial differential equations – innovations and applications, Volume 3 (eds Mendaldi, J. M., Rofman, E. and Sulem, A.), (IOS Press, Amsterdam, 2000) 1–16; https://core.ac.uk/download/pdf/30830946.pdf.Google Scholar
Pang, T. and Hussain, A., “An application of functional Itô’s formula to stochastic portfolio optimization with bounded memory”, in: SIAM Proceedings of the Conference on Control and its Applications, Paris, France (2015) 159–166; doi:10.1137/1.9781611974072.23.Google Scholar
Pang, T. and Hussain, A., “An infinite time horizon portfolio optimization model with delays”, Math. Control Relat. Fields 6 (2016) 629–651; doi:10.3934/mcrf.2016018.Google Scholar
Shen, Y., Meng, Q. and Shi, P., “Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance”, Automatica 50 (2014) 1565–1579; doi:10.1016/j.automatica.2014.03.021.Google Scholar
Shen, Y. and Zeng, Y., “Optimal investment–reinsurance strategy for mean-variance insurers: a maximum principle approach”, Insurance Math. Econom. 57 (2014) 1–12; doi:10.1016/j.insmatheco.2014.04.004.Google Scholar
Zariphopoulou, T., “Optimal investment and consumption models with non-linear stock dynamics”, Math. Methods Oper. Res. 50 (1999) 271–296; doi:10.1007/s001860050098.Google Scholar
Zariphopoulou, T., “A solution approach to valuation with unhedgeable risks”, Finance Stoch. 5 (2001) 61–82; doi:10.1007/PL00000040.Google Scholar
Zariphopoulou, T., “Optimal asset allocation in a stochastic factor model – an overview and open problems”, in: Advanced financial modeling, Volume 8, RADON Series on Computational and Applied Mathematics (eds Albrecher, H., Runggaldier, W. J. and Schachermayer, W.), (Walter de Gruyter, Berlin, 2009) 427–453; https://web.ma.utexas.edu/users/zariphop/pdfs/TZ-Submitted-11.pdf.Google Scholar