IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v126y2016i9p2615-2633.html
   My bibliography  Save this article

Muckenhoupt’s (Ap) condition and the existence of the optimal martingale measure

Author

Listed:
  • Kramkov, Dmitry
  • Weston, Kim

Abstract

In the problem of optimal investment with a utility function defined on (0,∞), we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt (Ap) condition for the power p=1/(1−a), where a∈(0,1) is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this (Ap) condition is sharp.

Suggested Citation

  • Kramkov, Dmitry & Weston, Kim, 2016. "Muckenhoupt’s (Ap) condition and the existence of the optimal martingale measure," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2615-2633.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:9:p:2615-2633
    DOI: 10.1016/j.spa.2016.02.012
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915300545
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2016.02.012?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Marcel Nutz, 2009. "The Opportunity Process for Optimal Consumption and Investment with Power Utility," Papers 0912.1879, arXiv.org, revised Jun 2010.
    2. He, Hua & Pearson, Neil D., 1991. "Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case," Journal of Economic Theory, Elsevier, vol. 54(2), pages 259-304, August.
    3. Hua He & Neil D. Pearson, 1991. "Consumption and Portfolio Policies With Incomplete Markets and Short‐Sale Constraints: the Finite‐Dimensional Case1," Mathematical Finance, Wiley Blackwell, vol. 1(3), pages 1-10, July.
    4. Kim, Tong Suk & Omberg, Edward, 1996. "Dynamic Nonmyopic Portfolio Behavior," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 141-161.
    5. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    6. Martin Schweizer & Christophe Stricker & Freddy Delbaen & Pascale Monat & Walter Schachermayer, 1997. "Weighted norm inequalities and hedging in incomplete markets," Finance and Stochastics, Springer, vol. 1(3), pages 181-227.
    7. 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.
    8. Mark Rubinstein, 1976. "The Valuation of Uncertain Income Streams and the Pricing of Options," Bell Journal of Economics, The RAND Corporation, vol. 7(2), pages 407-425, Autumn.
    9. Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123, April.
    10. Julien Hugonnier & Dmitry Kramkov & Walter Schachermayer, 2005. "On Utility‐Based Pricing Of Contingent Claims In Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 203-212, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Çetin, Umut & Larsen, Kasper, 2023. "Uniqueness in cauchy problems for diffusive real-valued strict local martingales," LSE Research Online Documents on Economics 118743, London School of Economics and Political Science, LSE Library.
    2. Kasper Larsen & Halil Mete Soner & Gordan Žitković, 2020. "Conditional Davis pricing," Finance and Stochastics, Springer, vol. 24(3), pages 565-599, July.

    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. Dmitry Kramkov & Kim Weston, 2015. "Muckenhoupt's $(A_p)$ condition and the existence of the optimal martingale measure," Papers 1507.05865, arXiv.org.
    2. Matoussi, Anis & Xing, Hao, 2018. "Convex duality for Epstein-Zin stochastic differential utility," LSE Research Online Documents on Economics 82519, London School of Economics and Political Science, LSE Library.
    3. Anis Matoussi & Hao Xing, 2016. "Convex duality for stochastic differential utility," Papers 1601.03562, arXiv.org.
    4. repec:dau:papers:123456789/5374 is not listed on IDEAS
    5. Bayraktar, Erhan & Kravitz, Ross, 2013. "Stability of exponential utility maximization with respect to market perturbations," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1671-1690.
    6. Marina Santacroce & Paola Siri & Barbara Trivellato, 2023. "Forward Backward SDEs Systems for Utility Maximization in Jump Diffusion Models," Papers 2302.08253, arXiv.org.
    7. Paolo Guasoni & Lóránt Nagy & Miklós Rásonyi, 2021. "Young, timid, and risk takers," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1332-1356, October.
    8. Thai Nguyen & Mitja Stadje, 2020. "Utility maximization under endogenous pricing," Papers 2005.04312, arXiv.org, revised Mar 2024.
    9. Julien Hugonnier & Dmitry Kramkov, 2004. "Optimal investment with random endowments in incomplete markets," Papers math/0405293, arXiv.org.
    10. Oleksii Mostovyi & Mihai Sîrbu, 2019. "Sensitivity analysis of the utility maximisation problem with respect to model perturbations," Finance and Stochastics, Springer, vol. 23(3), pages 595-640, July.
    11. Thijs Kamma & Antoon Pelsser, 2019. "Near-Optimal Dynamic Asset Allocation in Financial Markets with Trading Constraints," Papers 1906.12317, arXiv.org, revised Oct 2019.
    12. Schroder, Mark & Skiadas, Costis, 2005. "Lifetime consumption-portfolio choice under trading constraints, recursive preferences, and nontradeable income," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 1-30, January.
    13. Keita Owari, 2011. "A Note on Utility Maximization with Unbounded Random Endowment," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 18(1), pages 89-103, March.
    14. David German, 2010. "Overview of utility-based valuation," Papers 1003.5712, arXiv.org.
    15. Czichowsky, Christoph & Schachermayer, Walter, 2017. "Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion," LSE Research Online Documents on Economics 67689, London School of Economics and Political Science, LSE Library.
    16. Chenxu Li & O. Scaillet & Yiwen Shen, 2020. "Decomposition of Optimal Dynamic Portfolio Choice with Wealth-Dependent Utilities in Incomplete Markets," Swiss Finance Institute Research Paper Series 20-22, Swiss Finance Institute.
    17. Constantinos Kardaras & Gordan Zitkovic, 2007. "Stability of the utility maximization problem with random endowment in incomplete markets," Papers 0706.0482, arXiv.org, revised Mar 2010.
    18. Martin B. Haugh & Leonid Kogan & Jiang Wang, 2006. "Evaluating Portfolio Policies: A Duality Approach," Operations Research, INFORMS, vol. 54(3), pages 405-418, June.
    19. Katia Colaneri & Stefano Herzel & Marco Nicolosi, 2021. "The value of knowing the market price of risk," Annals of Operations Research, Springer, vol. 299(1), pages 101-131, April.
    20. Oleksii Mostovyi & Mihai S^irbu, 2017. "Sensitivity analysis of the utility maximization problem with respect to model perturbations," Papers 1705.08291, arXiv.org.
    21. Mahan Tahvildari, 2021. "Forward indifference valuation and hedging of basis risk under partial information," Papers 2101.00251, arXiv.org.

    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:eee:spapps:v:126:y:2016:i:9:p:2615-2633. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.