IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1308.2254.html
   My bibliography  Save this paper

Optimal investment for all time horizons and Martin boundary of space-time diffusions

Author

Listed:
  • Sergey Nadtochiy
  • Michael Tehranchi

Abstract

This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton-Jacobi-Bellman equation on a semi-infinite time interval. In the case when this equation can be linearized, the problem reduces to a time-reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additional uniform ellipticity condition, we make use of the available description of all minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all positive solutions. These results allow us to construct a large family of the aforementioned optimality criteria, including some closed form examples in relevant financial models.

Suggested Citation

  • Sergey Nadtochiy & Michael Tehranchi, 2013. "Optimal investment for all time horizons and Martin boundary of space-time diffusions," Papers 1308.2254, arXiv.org, revised Jan 2014.
  • Handle: RePEc:arx:papers:1308.2254
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1308.2254
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Cox, John C. & Huang, Chi-fu, 1992. "A continuous-time portfolio turnpike theorem," Journal of Economic Dynamics and Control, Elsevier, vol. 16(3-4), pages 491-507.
    2. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    3. Schwartz, Eduardo S, 1997. "The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance, American Finance Association, vol. 52(3), pages 923-973, July.
    4. Jérome Detemple & Marcel Rindisbacher, 2010. "Dynamic Asset Allocation: Portfolio Decomposition Formula and Applications," Review of Financial Studies, Society for Financial Studies, vol. 23(1), pages 25-100, January.
    5. Stanley R. Pliska, 1986. "A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 371-382, May.
    6. Paolo Guasoni & Scott Robertson, 2012. "Portfolios and risk premia for the long run," Papers 1203.1399, arXiv.org.
    7. Henderson, Vicky & Hobson, David, 2007. "Horizon-unbiased utility functions," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1621-1641, November.
    8. Karni, Edi & Schmeidler, David & Vind, Karl, 1983. "On State Dependent Preferences and Subjective Probabilities," Econometrica, Econometric Society, vol. 51(4), pages 1021-1031, July.
    9. Nicole El Karoui & Mohamed M'Rad, 2010. "Stochastic Utilities With a Given Optimal Portfolio : Approach by Stochastic Flows," Working Papers hal-00477380, HAL.
    10. Dmitry Kramkov & Mihai S^{{i}}rbu, 2006. "On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets," Papers math/0610224, arXiv.org.
    11. M. Musiela & T. Zariphopoulou, 2009. "Portfolio choice under dynamic investment performance criteria," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 161-170.
    12. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    13. MOSSIN, Jan, 1968. "Optimal multiperiod portfolio policies," LIDAM Reprints CORE 19, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Kreps, David M & Porteus, Evan L, 1978. "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica, Econometric Society, vol. 46(1), pages 185-200, January.
    15. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li, Minqiang, 2010. "Asset Pricing - A Brief Review," MPRA Paper 22379, University Library of Munich, Germany.
    2. Goncalo dos Reis & Vadim Platonov, 2020. "Forward utility and market adjustments in relative investment-consumption games of many players," Papers 2012.01235, arXiv.org, revised Mar 2022.
    3. Knut K. Aase & Petter Bjerksund, 2021. "The Optimal Spending Rate versus the Expected Real Return of a Sovereign Wealth Fund," JRFM, MDPI, vol. 14(9), pages 1-36, September.
    4. Paolo Guasoni & Constantinos Kardaras & Scott Robertson & Hao Xing, 2014. "Abstract, classic, and explicit turnpikes," Finance and Stochastics, Springer, vol. 18(1), pages 75-114, January.
    5. Jérôme Detemple, 2014. "Portfolio Selection: A Review," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 1-21, April.
    6. Sigrid Källblad & Jan Obłój & Thaleia Zariphopoulou, 2018. "Dynamically consistent investment under model uncertainty: the robust forward criteria," Finance and Stochastics, Springer, vol. 22(4), pages 879-918, October.
    7. Nicole El Karoui & Mohamed Mrad, 2013. "An Exact Connection between two Solvable SDEs and a Nonlinear Utility Stochastic PDE," Post-Print hal-00477381, HAL.
    8. Leonid Kogan & Raman Uppal, "undated". "Risk Aversion and Optimal Portfolio Policies in Partial and General Equilibrium Economies," Rodney L. White Center for Financial Research Working Papers 13-00, Wharton School Rodney L. White Center for Financial Research.
    9. Anis Matoussi & Hao Xing, 2016. "Convex duality for stochastic differential utility," Papers 1601.03562, arXiv.org.
    10. Schroder, Mark & Skiadas, Costis, 1999. "Optimal Consumption and Portfolio Selection with Stochastic Differential Utility," Journal of Economic Theory, Elsevier, vol. 89(1), pages 68-126, November.
    11. Hao Xing, 2015. "Consumption investment optimization with Epstein-Zin utility in incomplete markets," Papers 1501.04747, arXiv.org, revised Nov 2015.
    12. Scott Robertson & Hao Xing, 2014. "Long Term Optimal Investment in Matrix Valued Factor Models," Papers 1408.7010, arXiv.org.
    13. Aase, Knut K., 2014. "The Life Cycle Model with Recursive Utility: New insights on optimal consumption," Discussion Papers 2014/19, Norwegian School of Economics, Department of Business and Management Science, revised 16 Oct 2015.
    14. Hubar, Sylwia & Koulovatianos, Christos & Li, Jian, 2020. "The role of labor-income risk in household risk-taking," European Economic Review, Elsevier, vol. 129(C).
    15. Li, Tongtong & Wang, Shibo & Yang, Jinqiang, 2021. "Robust consumption and portfolio choices with habit formation," Economic Modelling, Elsevier, vol. 98(C), pages 227-246.
    16. Mahan Tahvildari, 2021. "Forward indifference valuation and hedging of basis risk under partial information," Papers 2101.00251, arXiv.org.
    17. Kraft, Holger & Munk, Claus & Weiss, Farina, 2022. "Bequest motives in consumption-portfolio decisions with recursive utility," Journal of Banking & Finance, Elsevier, vol. 138(C).
    18. Stephen G. Dimmock & Neng Wang & Jinqiang Yang, 2019. "The Endowment Model and Modern Portfolio Theory," NBER Working Papers 25559, National Bureau of Economic Research, Inc.
    19. Hao Xing, 2017. "Consumption–investment optimization with Epstein–Zin utility in incomplete markets," Finance and Stochastics, Springer, vol. 21(1), pages 227-262, January.
    20. John Armstrong & Cristin Buescu, 2020. "Asymptotically Optimal Management of Heterogeneous Collectivised Investment Funds," Papers 2004.01506, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1308.2254. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.