Finance and Stochastics

, Volume 18, Issue 1, pp 75–114 | Cite as

Abstract, classic, and explicit turnpikes

  • Paolo Guasoni
  • Constantinos Kardaras
  • Scott Robertson
  • Hao Xing


Portfolio turnpikes state that as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike states that optimal final payoffs and portfolios converge under their myopic probabilities. In diffusion models with several assets and a single state variable, the classic turnpike demonstrates that optimal portfolios converge under the physical probability. In the same setting, the explicit turnpike identifies the limit of finite-horizon optimal portfolios as a long-run myopic portfolio defined in terms of the solution of an ergodic HJB equation.


Portfolio choice Incomplete markets Long-run Utility functions Turnpikes 

Mathematics Subject Classification

93E20 60H30 

JEL Classification

G11 C61 



We are grateful to the Associate Editor and two anonymous referees for carefully reading this paper and providing valuable suggestions, which greatly assisted us in improving this paper.

Paolo Guasoni is partially supported by the ERC (278295), NSF (DMS-0807994, DMS-1109047), SFI (07/MI/008, 07/SK/M1189, 08/SRC/FMC1389), and FP7 (RG-248896). Constantinos Kardaras is partially supported by the NSF DMS-0908461. Hao Xing is partially supported by an LSE STICERD grant.


  1. 1.
    Benninga, S., Mayshar, J.: Heterogeneity and option pricing. Rev. Deriv. Res. 4, 7–27 (2000) CrossRefMATHGoogle Scholar
  2. 2.
    Bichteler, K.: Stochastic Integration with Jumps. Encyclopedia of Mathematics and Its Applications, vol. 89. Cambridge University Press, Cambridge (2002) CrossRefMATHGoogle Scholar
  3. 3.
    Bielecki, T.R., Pliska, S.R.: Risk sensitive asset management with transaction costs. Finance Stoch. 4, 1–33 (2000) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bielecki, T.R., Hernandez-Hernandez, D., Pliska, S.R.: Risk sensitive asset management with constrained trading strategies. In: Yong, J. (ed.) Recent Developments in Mathematical Finance, Shanghai, 2001, pp. 127–138. World Scientific, River Edge (2002) Google Scholar
  5. 5.
    Cheridito, P., Summer, C.: Utility maximization under increasing risk aversion in one-period models. Finance Stoch. 10, 147–158 (2006) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cheridito, P., Filipović, D., Yor, M.: Equivalent and absolutely continuous measure changes for jump-diffusion processes. Ann. Appl. Probab. 15, 1713–1732 (2005) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cox, J.C., Huang, C.: A continuous-time portfolio turnpike theorem. J. Econ. Dyn. Control 16, 491–507 (1992) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cvitanić, J., Malamud, S.: Price impact and portfolio impact. J. Financ. Econ. 100, 201–225 (2011) CrossRefGoogle Scholar
  9. 9.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 215–250 (1994) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 463–520 (1998) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Detemple, J., Rindisbacher, M.: Dynamic asset allocation: portfolio decomposition formula and applications. Rev. Financ. Stud. 23, 25–100 (2010) CrossRefGoogle Scholar
  12. 12.
    Dybvig, P.H., Rogers, L.C.G., Back, K.: Portfolio turnpikes. Rev. Financ. Stud. 12, 165–195 (1999) CrossRefGoogle Scholar
  13. 13.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998) MATHGoogle Scholar
  14. 14.
    Feller, W.: Two singular diffusion problems. Ann. Math. 54, 173–182 (1951) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Fleming, W.H., McEneaney, W.M.: Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim. 33, 1881–1915 (1995) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. Math. Finance 10, 197–213 (2000) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. II. Ann. Appl. Probab. 12, 730–767 (2002) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964) MATHGoogle Scholar
  19. 19.
    Friedman, A.: Stochastic Differential Equations and Applications. Vol. 1. Probability and Mathematical Statistics, vol. 28. Academic Press/Harcourt Brace Jovanovich, New York (1975) MATHGoogle Scholar
  20. 20.
    Goll, T., Kallsen, J.: A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13, 774–799 (2003) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Guasoni, P., Robertson, S.: Portfolios and risk premia for the long run. Ann. Appl. Probab. 22, 239–284 (2012) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Guasoni, P., Robertson, S.: Static fund separation of long-term investments (2013). Math. Financ. Available at
  23. 23.
    Hakansson, N.H.: Convergence to isoelastic utility and policy in multiperiod portfolio choice. J. Financ. Econ. 1, 201–224 (1974) CrossRefGoogle Scholar
  24. 24.
    Heath, D., Schweizer, M.: Martingales versus PDEs in finance: an equivalence result with examples. J. Appl. Probab. 37, 947–957 (2000) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Huang, C.-F., Zariphopoulou, T.: Turnpike behavior of long-term investments. Finance Stoch. 3, 15–34 (1999) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Huberman, G., Ross, S.: Portfolio turnpike theorems, risk aversion, and regularly varying utility functions. Econometrica 51, 1345–1361 (1983) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 288. Springer, Berlin (2003). [Fundamental Principles of Mathematical Sciences] CrossRefMATHGoogle Scholar
  28. 28.
    Jin, X.: Consumption and portfolio turnpike theorems in a continuous-time finance model. J. Econ. Dyn. Control 22, 1001–1026 (1998) CrossRefMATHGoogle Scholar
  29. 29.
    Kabanov, Y., Kramkov, D.: Asymptotic arbitrage in large financial markets. Finance Stoch. 2, 143–172 (1998) CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Kaise, H., Sheu, S.J.: On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control. Ann. Probab. 34, 284–320 (2006) CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Kallsen, J.: Optimal portfolios for exponential Lévy processes. Math. Methods Oper. Res. 51, 357–374 (2000) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493 (2007) CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kardaras, C.: The continuous behavior of the numéraire portfolio under small changes in information structure, probabilistic views and investment constraints. Stoch. Process. Appl. 120, 331–347 (2010) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Kramkov, D., Sîrbu, M.: On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 16, 1352–1384 (2006) CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Kramkov, D., Sîrbu, M.: Sensitivity analysis of utility-based prices and risk-tolerance wealth processes. Ann. Appl. Probab. 16, 2140–2194 (2006) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Kramkov, D., Sîrbu, M.: Asymptotic analysis of utility-based hedging strategies for small number of contingent claims. Stoch. Process. Appl. 117, 1606–1620 (2007) CrossRefMATHGoogle Scholar
  38. 38.
    Larsen, K., Žitković, G.: Stability of utility-maximization in incomplete markets. Stoch. Process. Appl. 117, 1642–1662 (2007) CrossRefMATHGoogle Scholar
  39. 39.
    Leland, H.: On turnpike portfolios. In: Szego, G., Shell, K. (eds.) Mathematical Methods in Investment and Finance, p. 24. North-Holland, Amsterdam (1972) Google Scholar
  40. 40.
    Mossin, J.: Optimal multiperiod portfolio policies. J. Bus. 41, 215–229 (1968) CrossRefGoogle Scholar
  41. 41.
    Nagai, H., Peng, S.: Risk-sensitive optimal investment problems with partial information on infinite time horizon. In: Yong, J. (ed.) Recent Developments in Mathematical Finance, Shanghai, 2001, pp. 85–98. World Scientific, River Edge (2002) Google Scholar
  42. 42.
    Nagai, H., Peng, S.: Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann. Appl. Probab. 12, 173–195 (2002) CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Najnudel, J., Nikeghbali, A.: A new kind of augmentation of filtrations. ESAIM Probab. Stat. 15, 39–57 (2011) CrossRefMathSciNetGoogle Scholar
  44. 44.
    Pinchover, Y.: Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators. J. Funct. Anal. 104, 54–70 (1992) CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Pinchover, Y.: Large time behavior of the heat kernel. J. Funct. Anal. 206, 191–209 (2004) CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995) CrossRefMATHGoogle Scholar
  47. 47.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus, Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000). Reprint of the second (1994) edition Google Scholar
  48. 48.
    Ross, S.: Portfolio turnpike theorems for constant policies. J. Financ. Econ. 1, 171–198 (1974) CrossRefGoogle Scholar
  49. 49.
    Zariphopoulou, T.: A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61–82 (2001) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Paolo Guasoni
    • 1
    • 2
  • Constantinos Kardaras
    • 3
  • Scott Robertson
    • 4
  • Hao Xing
    • 3
  1. 1.School of Mathematical SciencesDublin City UniversityDublin 9Ireland
  2. 2.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  3. 3.Department of StatisticsLondon School of Economics and Political ScienceLondonUK
  4. 4.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations