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Abstract, classic, and explicit turnpikes

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Abstract

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.

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Notes

  1. This interpretation underpins the literature on risk-sensitive control, introduced by Fleming and McEneaney [15], and applied to optimal portfolio choice by Bielecki et al. [4], Bielecki and Pliska [3], Fleming and Sheu [16, 17], Nagai and Peng [41, 42], among others.

  2. A subset A of Ω is N-negligible if there exists a sequence (B n ) n≥0 of subsets of Ω such that for all n≥0, we have \(B_{n} \in\mathcal{F}_{n}\) and \(\mathbb {P}[B_{n}]=0\), and A⊂⋃ n≥0 B n . This notion is introduced in [2, Definition 1.3.23] and [43]. Such a completion of \(\mathcal {F}_{0}\) ensures, for all T≥0, that the space \((\varOmega, \mathcal {F}_{T}, (\mathcal{F}_{t})_{0\leq t\leq T}, \mathbb {P})\) satisfies the usual conditions. Hence all references below on finite-horizon problems with completed filtration can be used in this paper.

  3. For any ϵ>0, there exists M ϵ such that U′(x)≤(1+ϵ)x p−1 for x≥M ϵ . Integrating the previous inequality on (M ϵ ,x) yields \(U(x) \leq(1+\epsilon)(x^{p}-M^{p}_{\varepsilon })/p + U(M_{\varepsilon })\), when x≥M ϵ and 0<p<1, from which the claim follows. The proof for the case p=0 is similar.

  4. These probabilities already appear in the work of Kramkov and Sîrbu [35–37] under the name of \(\bf R\).

  5. The notation \(\widetilde{\mathbb {Q}}\)-lim T→∞ is short for the limit in probability under \(\widetilde{\mathbb {Q}}\).

  6. Since R 0=0 by assumption, \(\mathbb {P}^{\xi}\) with ξ=(0,y) is denoted as \(\mathbb {P}^{y}\). The same convention applies to other probabilities introduced later.

  7. In the model (2.7) and (2.8), u 0,T depends on the initial value of the state variable Y 0=y. Hence u T is a function of y. Since Proposition 2.5 reduces the problem to the comparison of the optimal isoelastic finite-horizon portfolio with its long-run limit, the superscript 0 will be omitted in this section.

  8. Any y 0∈E suffices. This y 0 is chosen to align m with \(\hat{m}\).

  9. If a logarithmic investor is present (γ i =1 for some i), a constant is added to U(x), and the stated equivalence remains valid.

References

  1. Benninga, S., Mayshar, J.: Heterogeneity and option pricing. Rev. Deriv. Res. 4, 7–27 (2000)

    Article  MATH  Google Scholar 

  2. Bichteler, K.: Stochastic Integration with Jumps. Encyclopedia of Mathematics and Its Applications, vol. 89. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  3. Bielecki, T.R., Pliska, S.R.: Risk sensitive asset management with transaction costs. Finance Stoch. 4, 1–33 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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. Cheridito, P., Summer, C.: Utility maximization under increasing risk aversion in one-period models. Finance Stoch. 10, 147–158 (2006)

    Article  MathSciNet  Google Scholar 

  6. Cheridito, P., Filipović, D., Yor, M.: Equivalent and absolutely continuous measure changes for jump-diffusion processes. Ann. Appl. Probab. 15, 1713–1732 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cox, J.C., Huang, C.: A continuous-time portfolio turnpike theorem. J. Econ. Dyn. Control 16, 491–507 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cvitanić, J., Malamud, S.: Price impact and portfolio impact. J. Financ. Econ. 100, 201–225 (2011)

    Article  Google Scholar 

  9. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 215–250 (1994)

    Article  MathSciNet  Google Scholar 

  10. Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 463–520 (1998)

    Article  MathSciNet  Google Scholar 

  11. Detemple, J., Rindisbacher, M.: Dynamic asset allocation: portfolio decomposition formula and applications. Rev. Financ. Stud. 23, 25–100 (2010)

    Article  Google Scholar 

  12. Dybvig, P.H., Rogers, L.C.G., Back, K.: Portfolio turnpikes. Rev. Financ. Stud. 12, 165–195 (1999)

    Article  Google Scholar 

  13. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  14. Feller, W.: Two singular diffusion problems. Ann. Math. 54, 173–182 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fleming, W.H., McEneaney, W.M.: Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim. 33, 1881–1915 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. Math. Finance 10, 197–213 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. II. Ann. Appl. Probab. 12, 730–767 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  19. Friedman, A.: Stochastic Differential Equations and Applications. Vol. 1. Probability and Mathematical Statistics, vol. 28. Academic Press/Harcourt Brace Jovanovich, New York (1975)

    MATH  Google Scholar 

  20. Goll, T., Kallsen, J.: A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13, 774–799 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Guasoni, P., Robertson, S.: Portfolios and risk premia for the long run. Ann. Appl. Probab. 22, 239–284 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Guasoni, P., Robertson, S.: Static fund separation of long-term investments (2013). Math. Financ. Available at http://onlinelibrary.wiley.com/doi/10.1111/mafi.12017/abstract

  23. Hakansson, N.H.: Convergence to isoelastic utility and policy in multiperiod portfolio choice. J. Financ. Econ. 1, 201–224 (1974)

    Article  Google Scholar 

  24. Heath, D., Schweizer, M.: Martingales versus PDEs in finance: an equivalence result with examples. J. Appl. Probab. 37, 947–957 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Huang, C.-F., Zariphopoulou, T.: Turnpike behavior of long-term investments. Finance Stoch. 3, 15–34 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Huberman, G., Ross, S.: Portfolio turnpike theorems, risk aversion, and regularly varying utility functions. Econometrica 51, 1345–1361 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  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]

    Book  MATH  Google Scholar 

  28. Jin, X.: Consumption and portfolio turnpike theorems in a continuous-time finance model. J. Econ. Dyn. Control 22, 1001–1026 (1998)

    Article  MATH  Google Scholar 

  29. Kabanov, Y., Kramkov, D.: Asymptotic arbitrage in large financial markets. Finance Stoch. 2, 143–172 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kallsen, J.: Optimal portfolios for exponential Lévy processes. Math. Methods Oper. Res. 51, 357–374 (2000)

    Article  MathSciNet  Google Scholar 

  32. Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kramkov, D., Sîrbu, M.: Sensitivity analysis of utility-based prices and risk-tolerance wealth processes. Ann. Appl. Probab. 16, 2140–2194 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  38. Larsen, K., Žitković, G.: Stability of utility-maximization in incomplete markets. Stoch. Process. Appl. 117, 1642–1662 (2007)

    Article  MATH  Google Scholar 

  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. Mossin, J.: Optimal multiperiod portfolio policies. J. Bus. 41, 215–229 (1968)

    Article  Google Scholar 

  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. Nagai, H., Peng, S.: Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann. Appl. Probab. 12, 173–195 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  43. Najnudel, J., Nikeghbali, A.: A new kind of augmentation of filtrations. ESAIM Probab. Stat. 15, 39–57 (2011)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  45. Pinchover, Y.: Large time behavior of the heat kernel. J. Funct. Anal. 206, 191–209 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  46. Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  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. Ross, S.: Portfolio turnpike theorems for constant policies. J. Financ. Econ. 1, 171–198 (1974)

    Article  Google Scholar 

  49. Zariphopoulou, T.: A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61–82 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

    Paolo Guasoni

  2. Department of Mathematics and Statistics, Boston University, 111 Cummington St, Boston, MA, 02215, USA

    Paolo Guasoni

  3. Department of Statistics, London School of Economics and Political Science, 10 Houghton St, London, WC2A 2AE, UK

    Constantinos Kardaras & Hao Xing

  4. Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall 6113, Pittsburgh, PA, 15213, USA

    Scott Robertson

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Guasoni, P., Kardaras, C., Robertson, S. et al. Abstract, classic, and explicit turnpikes. Finance Stoch 18, 75–114 (2014). https://doi.org/10.1007/s00780-013-0216-5

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