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

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  • Sergey Nadtochiy
  • Michael Tehranchi
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    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.

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    File URL: http://arxiv.org/pdf/1308.2254
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1308.2254.

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    Date of creation: Aug 2013
    Date of revision: Jan 2014
    Handle: RePEc:arx:papers:1308.2254

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    Web page: http://arxiv.org/

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    1. Schwartz, Eduardo S, 1997. " The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance, American Finance Association, American Finance Association, vol. 52(3), pages 923-73, July.
    2. Paolo Guasoni & Scott Robertson, 2012. "Portfolios and risk premia for the long run," Papers 1203.1399, arXiv.org.
    3. Karni, Edi & Schmeidler, David & Vind, Karl, 1983. "On State Dependent Preferences and Subjective Probabilities," Econometrica, Econometric Society, Econometric Society, vol. 51(4), pages 1021-31, July.
    4. Cox, John C. & Huang, Chi-fu, 1992. "A continuous-time portfolio turnpike theorem," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 16(3-4), pages 491-507.
    5. M. Musiela & T. Zariphopoulou, 2009. "Portfolio choice under dynamic investment performance criteria," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 9(2), pages 161-170.
    6. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-36.
    7. Kreps, David M & Porteus, Evan L, 1978. "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica, Econometric Society, Econometric Society, vol. 46(1), pages 185-200, January.
    8. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, Econometric Society, vol. 60(2), pages 353-94, March.
    9. Jérome Detemple & Marcel Rindisbacher, 2010. "Dynamic Asset Allocation: Portfolio Decomposition Formula and Applications," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 23(1), pages 25-100, January.
    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.
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