Skip to main content Accessibility help
×
×
Home

OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY

  • L. LI (a1) and H. MI (a1)
Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      OPTIMAL INVESTMENT AND CONSUMPTION WITH STOCHASTIC FACTOR AND DELAY
      Available formats
      ×
Copyright
Corresponding author
References
Hide All
[1] 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) 181196; doi:10.1016/j.insmatheco.2015.01.005.
[2] C.-X. Aand Shao, Y., “Portfolio optimization problem with delay under Cox–Ingersoll–Ross model”, J. Math. Finance 7 (2017) 699717; doi:10.4236/jmf.2017.73037.
[3] Agram, N., Haadem, S., Øksendal, B. and Proske, F., “A maximum principle for infinite horizon delay equations”, SIAM J. Math. Anal. 45 (2013) 24992522; doi:10.1137/120882809.
[4] Bielecki, T. R. and Pliska, S. R., “Risk-sensitive dynamic asset management”, Appl. Math. Optim. 39 (1999) 337360; doi:10.1007/s002459900110.
[5] Chacko, G. and Viceira, L. M., “Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets”, Rev. Financ. Stud. 18 (2005) 13691402; doi:10.1093/rfs/hhi035.
[6] Chang, M.-H., Pang, T. and Yang, Y., “A stochastic portfolio optimization model with bounded memory”, Math. Oper. Res. 36 (2011) 604619; doi:10.1287/moor.1110.0508.
[7] Cox, J. C. and Huang, C. F., “Optimal consumption and portfolio policies when asset prices follow a diffusion process”, J. Econom. Theory 49 (1989) 3383; doi:10.1016/0022-0531(89)90067-7.
[8] Delong, Ł. and Klüppelberg, C., “Optimal investment and consumption in a Black–Scholes market with Lévy-driven stochastic coefficients”, Ann. Appl. Probab. 18 (2008) 879908; doi:10.1214/07-AAP475.
[9] Elsanosi, I., Øksendal, B. and Sulem, A., “Some solvable stochastic control problems with delay”, Stochastics 71 (2000) 6989; doi:10.1080/17442500008834259.
[10] Fleming, W. H. and Hernández-Hernández, D., “An optimal consumption model with stochastic volatility”, Finance Stoch. 7 (2003) 245262; doi:10.1007/s0078002000.
[11] Fouque, J.-P., Papanicolaou, G. and Sircar, K. R., Derivatives in financial markets with stochastic volatility (Cambridge University Press, Cambridge, 2000).
[12] Hernández-Hernández, D. and Schied, A., “Robust utility maximization in a stochastic factor model”, Statist. Decisions 24 (2006) 109125; doi:10.1524/stnd.2006.24.1.109.
[13] Liu, J., “Portfolio selection in stochastic environments”, Rev. Financ. Stud. 20 (2007) 139; doi:10.1093/rfs/hhl001.
[14] Liu, J. and Pan, J., “Dynamic derivative strategies”, J. Financ. Econ. 69 (2003) 401430; doi:10.1016/S0304-405X(03)00118-1.
[15] Merton, R. C., “Lifetime portfolio selection under uncertainty: the continuous-time case”, Rev. Econ. Stat. 51 (1969) 247257; doi:10.2307/1926560.
[16] Merton, R. C., “Optimum consumption and portfolio rules in a continuous-time model”, J. Econom. Theory 3 (1971) 373413; doi:10.1016/0022-0531(71)90038-X.
[17] 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) 116; https://core.ac.uk/download/pdf/30830946.pdf.
[18] 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) 159166; doi:10.1137/1.9781611974072.23.
[19] Pang, T. and Hussain, A., “An infinite time horizon portfolio optimization model with delays”, Math. Control Relat. Fields 6 (2016) 629651; doi:10.3934/mcrf.2016018.
[20] 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) 15651579; doi:10.1016/j.automatica.2014.03.021.
[21] Shen, Y. and Zeng, Y., “Optimal investment–reinsurance strategy for mean-variance insurers: a maximum principle approach”, Insurance Math. Econom. 57 (2014) 112; doi:10.1016/j.insmatheco.2014.04.004.
[22] Zariphopoulou, T., “Optimal investment and consumption models with non-linear stock dynamics”, Math. Methods Oper. Res. 50 (1999) 271296; doi:10.1007/s001860050098.
[23] Zariphopoulou, T., “A solution approach to valuation with unhedgeable risks”, Finance Stoch. 5 (2001) 6182; doi:10.1007/PL00000040.
[24] 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) 427453; https://web.ma.utexas.edu/users/zariphop/pdfs/TZ-Submitted-11.pdf.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed