Mathematics and Financial Economics

, Volume 13, Issue 2, pp 227–251 | Cite as

Turnpike property and convergence rate for an investment and consumption model

  • Baojun Bian
  • Harry ZhengEmail author


We discuss the turnpike property for optimal investment and consumption problems. We find there exists a threshold value that determines the turnpike property for investment policy. The threshold value only depends on the Sharpe ratio, the riskless interest rate and the discount rate. We show that if utilities behave asymptotically like power utilities and satisfy some simple relations with the threshold value, then the turnpike property for investment holds. There is in general no turnpike property for consumption policy. We also provide the rate of convergence and illustrate the main results with examples of power and non-HARA utilities and numerical tests.


Optimal investment and consumption Turnpike property Convergence rate Dual control method 

JEL Classification

D9 G1 



The authors are grateful to the anonymous reviewer whose comments and suggestions have helped to improve the previous two versions.


  1. 1.
    Back, K., Dybvig, P.H., Rogers, L.C.G.: Portfolio turnpikes. Rev. Financial Stud. 12, 165–195 (1999)CrossRefGoogle Scholar
  2. 2.
    Bian, B., Miao, S., Zheng, H.: Smooth value functions for a class of nonsmooth utility maximization problems. SIAM J. Financial Math. 2, 727–747 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bian, B., Zheng, H.: Turnpike property and convergence rate for an investment model with general utility functions. J. Econ. Dyn. Control 51, 28–49 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cox, J., Huang, C.: A continuous time portfolio turnpike theorem. J. Econ. Dyn. Control 2, 491–507 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guasoni, P., Kardaras, C., Robertson, S., Xing, H.: Abstract, classic, and explicit turnpikes. Finance Stoch. 18, 75–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huang, C., Zariphopoulou, T.: Turnpike behaviour of long-term investments. Finance Stoch. 3, 15–34 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Huberman, G., Ross, S.: Portfolio turnpike theorems, risk aversion, and regularly varying functions. Econometrica 51, 1345–1361 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jin, X.: Consumption and portfolio turnpike theorems in a continuous-time finance model. J. Econ. Dyn. Control 22, 1001–1026 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Robertson, S., Xing, H.: Long term optimal investment in matrix valued factor models. SIAM J. Financial Math. 8, 400–434 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiChina
  2. 2.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations