Nearly-Optimal Asset Allocation in Hybrid Stock Investment Models

  • Q. Zhang
  • G. Yin


This work develops a class of stock-investment models that are hybrid in nature and involve continuous dynamics and discrete-event interventions. In lieu of the usual geometric Brownian motion formulation, hybrid geometric Brownian motion models are proposed, in which both the expected return and the volatility depend on a finite-state Markov chain. Our objective is to find nearly-optimal asset allocation strategies so as to maximize the expected returns. The use of the Markov chain stems from the motivation of capturing the market trends as well as various economic factors. To incorporate these economic factors into the models, the underlying Markov chain inevitably has a large state space. To reduce the complexity, a hierarchical approach is suggested, which leads to singularly-perturbed switching diffusion processes. By aggregating the states of the Markov chains in each weakly irreducible class into a single state, limit switching diffusion processes are obtained. Using such asymptotic properties, nearly-optimal asset allocation policies are developed.

Markov chains switching diffusions hybrid models weak convergence near optimality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Duffie, D., Dynamic Asset Pricing Theory, 2nd Edition, Princeton University Press, Princeton, New Jersey, 1996.Google Scholar
  2. 2.
    Fougue, J. P., Papanicolaou, G., and Sircar, K. R., Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, England, 2000.Google Scholar
  3. 3.
    Hull, J. C., Options, Futures, and Other Derivatives, 3rd Edition, Prentice Hall, Upper Saddle River, New Jersey, 1997.Google Scholar
  4. 4.
    Karatzas, I., and Shreve, S. E., Methods of Mathematical Finance, Springer, New York, NY, 1998.Google Scholar
  5. 5.
    Musiela, M., and Rutkowski, M., Martingale Methods in Financial Modeling, Springer, New York, NY, 1997.Google Scholar
  6. 6.
    Merton, R. C., Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case, Review of Economics and Statistics, Vol. 51, pp. 247–257, 1969.Google Scholar
  7. 7.
    Zariphopoulou, T., Investment-Consumption Models with Transactions Costs and Markov Chain Parameters, SIAM Journal on Control and Optimization, Vol. 30, pp. 613–636, 1992.Google Scholar
  8. 8.
    Zhang, Q., Stock Trading: An Optimal Selling Rule , SIAM Journal on Control and Optimization, Vol. 40, pp. 64–87, 2001.Google Scholar
  9. 9.
    Yin, G., and Zhang, Q., Continuous-Time Markov Chains and Applications: A Singular-Perturbation Approach, Springer Verlag, New York, NY, 1998.Google Scholar
  10. 10.
    Yin, G., Zhang, Q., and Badowski, G., Asymptotic Properties of a Singularly-Perturbed Markov Chain with Inclusion of Transient States, Annals of Applied Probability, Vol. 10, pp. 549–572, 2000.Google Scholar
  11. 11.
    Billingsley, P., Convergence of Probability Measures, Wiley, New York, NY, 1968.Google Scholar
  12. 12.
    Ethier, S. N., and Kurtz, T. G., Markov Processes: Characterization and Convergence, Wiley, New York, NY, 1986.Google Scholar
  13. 13.
    Kushner, H. J., Approximation and Weak Convergence Methods for Randomn Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, Massachusetts, 1984.Google Scholar
  14. 14.
    Fleming, W. H., and Soner, H. M., Controlled Markov Processes and Viscosity Solutions, Springer Verlag, New York, NY, 1992.Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Q. Zhang
    • 1
  • G. Yin
    • 2
  1. 1.Department of MathematicsUniversity of GeorgiaAthensGeorgia
  2. 2.Department of MathematicsWayne State UniversityDetroit

Personalised recommendations