IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2007.15041.html
   My bibliography  Save this paper

Uniqueness in Cauchy problems for diffusive real-valued strict local martingales

Author

Listed:
  • Umut Cetin
  • Kasper Larsen

Abstract

For a real-valued one dimensional diffusive strict local martingale,, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution under a local H\"older condition. Under the weaker Engelbert-Schmidt conditions, we provide a set in which the Cauchy problem has a unique weak solution. We exemplify our results using quadratic normal volatility models and the two dimensional Bessel process.

Suggested Citation

  • Umut Cetin & Kasper Larsen, 2020. "Uniqueness in Cauchy problems for diffusive real-valued strict local martingales," Papers 2007.15041, arXiv.org, revised May 2022.
    • Handle: RePEc:arx:papers:2007.15041
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2007.15041
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Pal, Soumik & Protter, Philip, 2010. "Analysis of continuous strict local martingales via h-transforms," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1424-1443, August.
    2. Basak, Suleyman & Cuoco, Domenico, 1998. "An Equilibrium Model with Restricted Stock Market Participation," The Review of Financial Studies, Society for Financial Studies, vol. 11(2), pages 309-341.
    3. Mark Loewenstein & Gregory A. Willard, 2000. "Local martingales, arbitrage, and viability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 16(1), pages 135-161.
    4. Chabakauri, Georgy, 2015. "Asset pricing with heterogeneous preferences, beliefs, and portfolio constraints," Journal of Monetary Economics, Elsevier, vol. 75(C), pages 21-34.
    5. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "Why are quadratic normal volatility models analytically tractable?," Papers 1202.6187, arXiv.org, revised Mar 2013.
    6. Kardaras, Constantinos & Kreher, Dörte & Nikeghbali, Ashkan, 2015. "Strict local martingales and bubbles," LSE Research Online Documents on Economics 64967, London School of Economics and Political Science, LSE Library.
    7. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.
    8. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
    9. Kardaras, Constantinos & Ruf, Johannes, 2019. "Projections of scaled bessel processes," LSE Research Online Documents on Economics 100939, London School of Economics and Political Science, LSE Library.
    10. Chabakauri, Georgy, 2015. "Asset pricing with heterogeneous preferences, beliefs, and portfolio constraints," LSE Research Online Documents on Economics 60810, London School of Economics and Political Science, LSE Library.
    11. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    12. Constantinos Kardaras & Johannes Ruf, 2020. "Filtration shrinkage, the structure of deflators, and failure of market completeness," Finance and Stochastics, Springer, vol. 24(4), pages 871-901, October.
    13. Hugonnier, Julien, 2012. "Rational asset pricing bubbles and portfolio constraints," Journal of Economic Theory, Elsevier, vol. 147(6), pages 2260-2302.
    14. Soumik Pal & Philip Protter, 2007. "Analysis of continuous strict local martingales via h-transforms," Papers 0711.1136, arXiv.org, revised Jun 2010.
    15. Kasper Larsen & Halil Soner & Gordan Žitković, 2016. "Facelifting in utility maximization," Finance and Stochastics, Springer, vol. 20(1), pages 99-121, January.
    16. Hardy Hulley & Johannes Ruf, 2019. "Weak Tail Conditions for Local Martingales," Published Paper Series 2019-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    17. Constantinos Kardaras & Johannes Ruf, 2019. "Filtration shrinkage, the structure of deflators, and failure of market completeness," Papers 1912.04652, arXiv.org, revised Aug 2020.
    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. 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. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, vol. 18(1), pages 115-144, January.
    3. Antoine Jacquier & Martin Keller-Ressel, 2015. "Implied volatility in strict local martingale models," Papers 1508.04351, arXiv.org.
    4. Paolo Guasoni & Miklós Rásonyi, 2015. "Fragility of arbitrage and bubbles in local martingale diffusion models," Finance and Stochastics, Springer, vol. 19(2), pages 215-231, April.
    5. Martin Herdegen & Dörte Kreher, 2022. "Bubbles in discrete-time models," Finance and Stochastics, Springer, vol. 26(4), pages 899-925, October.
    6. Yukihiro Tsuzuki, 2024. "Boundary conditions at infinity for Black-Scholes equations," Papers 2401.05549, arXiv.org, revised Sep 2024.
    7. Johannes Muhle-Karbe & Marcel Nutz, 2018. "A risk-neutral equilibrium leading to uncertain volatility pricing," Finance and Stochastics, Springer, vol. 22(2), pages 281-295, April.
    8. Li, Xue-Mei, 2017. "Strict local martingales: Examples," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 65-68.
    9. Dirk Veestraeten, 2017. "On the multiplicity of option prices under CEV with positive elasticity of variance," Review of Derivatives Research, Springer, vol. 20(1), pages 1-13, April.
    10. Robert A. Jarrow, 2015. "Asset Price Bubbles," Annual Review of Financial Economics, Annual Reviews, vol. 7(1), pages 201-218, December.
    11. Kim Weston, 2022. "Existence of an equilibrium with limited participation," Papers 2206.12399, arXiv.org.
    12. Constantinos Kardaras, 2012. "Valuation and parities for exchange options," Papers 1206.3220, arXiv.org, revised Nov 2014.
    13. Petteri Piiroinen & Lassi Roininen & Tobias Schoden & Martin Simon, 2018. "Asset Price Bubbles: An Option-based Indicator," Papers 1805.07403, arXiv.org, revised Jul 2018.
    14. Hugonnier, Julien & Prieto, Rodolfo, 2015. "Asset pricing with arbitrage activity," Journal of Financial Economics, Elsevier, vol. 115(2), pages 411-428.
    15. Jarrow, Robert & Protter, Philip, 2012. "Discrete versus continuous time models: Local martingales and singular processes in asset pricing theory," Finance Research Letters, Elsevier, vol. 9(2), pages 58-62.
    16. Cetin, Umut, 2018. "Diffusion transformations, Black-Scholes equation and optimal stopping," LSE Research Online Documents on Economics 87261, London School of Economics and Political Science, LSE Library.
    17. Aleksandar Mijatovic & Mikhail Urusov, 2009. "On the Martingale Property of Certain Local Martingales," Papers 0905.3701, arXiv.org, revised Oct 2010.
    18. Baldeaux, Jan & Ignatieva, Katja & Platen, Eckhard, 2018. "Detecting money market bubbles," Journal of Banking & Finance, Elsevier, vol. 87(C), pages 369-379.
    19. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2012. "Strict local martingale deflators and valuing American call-type options," Finance and Stochastics, Springer, vol. 16(2), pages 275-291, April.
    20. José Carlos Dias & João Pedro Vidal Nunes & Aricson Cruz, 2020. "A note on options and bubbles under the CEV model: implications for pricing and hedging," Review of Derivatives Research, Springer, vol. 23(3), pages 249-272, October.

    More about this item

    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:2007.15041. 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.