IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v34y2021i1d10.1007_s10959-019-00969-y.html
   My bibliography  Save this article

Existence and Uniqueness Results for Time-Inhomogeneous Time-Change Equations and Fokker–Planck Equations

Author

Listed:
  • Leif Döring

    (University of Mannheim)

  • Lukas Gonon

    (University of St. Gallen)

  • David J. Prömel

    (University of Oxford)

  • Oleg Reichmann

    (European Investment Bank)

Abstract

We prove existence and uniqueness of solutions to Fokker–Planck equations associated with Markov operators multiplicatively perturbed by degenerate time-inhomogeneous coefficients. Precise conditions on the time-inhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be either globally bounded or bounded away from zero. The approach is based on constructing random time-changes and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.

Suggested Citation

  • Leif Döring & Lukas Gonon & David J. Prömel & Oleg Reichmann, 2021. "Existence and Uniqueness Results for Time-Inhomogeneous Time-Change Equations and Fokker–Planck Equations," Journal of Theoretical Probability, Springer, vol. 34(1), pages 173-205, March.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-019-00969-y
    DOI: 10.1007/s10959-019-00969-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-019-00969-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-019-00969-y?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. Peter Carr & Helyette Geman & Dilip Madan & Marc Yor, 2004. "From local volatility to local Levy models," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 581-588.
    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. Peter K. Friz & Stefan Gerhold & Marc Yor, 2013. "How to make Dupire's local volatility work with jumps," Papers 1302.5548, arXiv.org.
    2. Xu, Guoping & Zheng, Harry, 2010. "Basket options valuation for a local volatility jump-diffusion model with the asymptotic expansion method," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 415-422, December.
    3. Carol Alexander & Leonardo Nogueira, 2007. "Model-free price hedge ratios for homogeneous claims on tradable assets," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 473-479.
    4. Genin, Adrien & Tankov, Peter, 2020. "Optimal importance sampling for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 20-46.
    5. Hainaut, Donatien & Leonenko, Nikolai, 2020. "Option pricing in illiquid markets: a fractional jump-diffusion approach," LIDAM Discussion Papers ISBA 2020003, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Adrien Genin & Peter Tankov, 2016. "Optimal importance sampling for L\'evy Processes," Papers 1608.04621, arXiv.org.
    7. Dilip B. Madan & Martijn Pistorius & Wim Schoutens, 2013. "The valuation of structured products using Markov chain models," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 125-136, January.
    8. Ernst Eberlein & Dilip Madan, 2009. "Sato processes and the valuation of structured products," Quantitative Finance, Taylor & Francis Journals, vol. 9(1), pages 27-42.
    9. Julien Claisse & Gaoyue Guo & Pierre Henry-Labordère, 2018. "Some Results on Skorokhod Embedding and Robust Hedging with Local Time," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 569-597, November.
    10. F. Antonelli & A. Ramponi & S. Scarlatti, 2016. "Random Time Forward-Starting Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(08), pages 1-25, December.
    11. Samuel N. Cohen & Lukasz Szpruch, 2011. "On Markovian solutions to Markov Chain BSDEs," Papers 1111.5739, arXiv.org.
    12. David Heath & Eckhard Platen, 2006. "Local volatility function models under a benchmark approach," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 197-206.
    13. Pierre Henry-Labordere & Nizar Touzi, 2013. "An Explicit Martingale Version of Brenier's Theorem," Papers 1302.4854, arXiv.org, revised Apr 2013.
    14. Amel Bentata & Rama Cont, 2015. "Forward equations for option prices in semimartingale models," Finance and Stochastics, Springer, vol. 19(3), pages 617-651, July.
    15. Eckhard Platen & Renata Rendek, 2012. "The Affine Nature of Aggregate Wealth Dynamics," Research Paper Series 322, Quantitative Finance Research Centre, University of Technology, Sydney.
    16. Julien Claisse & Gaoyue Guo & Pierre Henry-Labordere, 2015. "Some Results on Skorokhod Embedding and Robust Hedging with Local Time," Papers 1511.07230, arXiv.org, revised Oct 2017.
    17. Rene Carmona & Yi Ma & Sergey Nadtochiy, 2015. "Simulation of Implied Volatility Surfaces via Tangent Levy Models," Papers 1504.00334, arXiv.org.
    18. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    19. Martin Schweizer & Johannes Wissel, 2008. "Arbitrage-free market models for option prices: the multi-strike case," Finance and Stochastics, Springer, vol. 12(4), pages 469-505, October.
    20. Peter Friz & Stefan Gerhold, 2011. "Don't stay local - extrapolation analytics for Dupire's local volatility," Papers 1105.1267, 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:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-019-00969-y. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.