IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v141y2021icp376-411.html
   My bibliography  Save this article

Locally Lipschitz BSDE driven by a continuous martingale a path-derivative approach

Author

Listed:
  • Nam, Kihun

Abstract

Using a new notion of path-derivative, we study the well-posedness of backward stochastic differential equation driven by a continuous martingale M when f(s,γ,y,z) is locally Lipschitz in (y,z): Yt=ξ(M[0,T])+∫tTf(s,M[0,s],Ys−,Zsms)dtr[M,M]s−∫tTZsdMs−NT+Nt.Here, M[0,t] is the path of M from 0 to t and m is defined by [M,M]t=∫0tmsms∗dtr[M,M]s. When the BSDE is one-dimensional, we show the existence and uniqueness of the solution. On the contrary, when the BSDE is multidimensional, we show the existence and uniqueness only when [M,M]T is small enough: otherwise, we provide a counterexample. Then, we investigate the applications to optimal control of diffusion and optimal portfolio selection under various restrictions.

Suggested Citation

  • Nam, Kihun, 2021. "Locally Lipschitz BSDE driven by a continuous martingale a path-derivative approach," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 376-411.
    • Handle: RePEc:eee:spapps:v:141:y:2021:i:c:p:376-411
    DOI: 10.1016/j.spa.2021.09.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414921001514
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2021.09.009?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. repec:dau:papers:123456789/7119 is not listed on IDEAS
    2. Hu, Ying & Tang, Shanjian, 2016. "Multi-dimensional backward stochastic differential equations of diagonally quadratic generators," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1066-1086.
    3. Marie-Amélie Morlais, 2009. "Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem," Finance and Stochastics, Springer, vol. 13(1), pages 121-150, January.
    4. Peter Imkeller & Anthony R'eveillac & Anja Richter, 2009. "Differentiability of quadratic BSDEs generated by continuous martingales," Papers 0907.0941, arXiv.org, revised Mar 2012.
    5. Briand, Philippe & Elie, Romuald, 2013. "A simple constructive approach to quadratic BSDEs with or without delay," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2921-2939.
    6. Patrick Cheridito & Ulrich Horst & Michael Kupper & Traian A. Pirvu, 2016. "Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 174-195, February.
    7. Monique Jeanblanc & Thibaut Mastrolia & Dylan Possamaï & Anthony Réveillac, 2015. "Utility Maximization With Random Horizon: A Bsde Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(07), pages 1-43, November.
    8. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    9. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    10. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Nov 2018.
    11. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
    12. Kupper, Michael & Luo, Peng & Tangpi, Ludovic, 2019. "Multidimensional Markovian FBSDEs with super-quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 902-923.
    13. Xing, Hao & Žitković, Gordan, 2018. "A class of globally solvable Markovian quadratic BSDE systems and applications," LSE Research Online Documents on Economics 73440, London School of Economics and Political Science, LSE Library.
    14. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    15. Tevzadze, Revaz, 2008. "Solvability of backward stochastic differential equations with quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 118(3), pages 503-515, March.
    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. Hu, Ying & Tang, Shanjian & Wang, Falei, 2022. "Quadratic G-BSDEs with convex generators and unbounded terminal conditions," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 363-390.
    2. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Nov 2018.
    3. Luis Escauriaza & Daniel C. Schwarz & Hao Xing, 2020. "Radner equilibrium and systems of quadratic BSDEs with discontinuous generators," Papers 2008.03500, arXiv.org, revised May 2021.
    4. Lionnet, Arnaud, 2014. "Some results on general quadratic reflected BSDEs driven by a continuous martingale," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1275-1302.
    5. Kim Weston, 2022. "Existence of an equilibrium with limited participation," Papers 2206.12399, arXiv.org.
    6. Dejian Tian, 2022. "Pricing principle via Tsallis relative entropy in incomplete market," Papers 2201.05316, arXiv.org, revised Oct 2022.
    7. Hernández, Camilo, 2023. "On quadratic multidimensional type-I BSVIEs, infinite families of BSDEs and their applications," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 249-298.
    8. Xing, Hao & Žitković, Gordan, 2018. "A class of globally solvable Markovian quadratic BSDE systems and applications," LSE Research Online Documents on Economics 73440, London School of Economics and Political Science, LSE Library.
    9. Kihun Nam, 2019. "Global Well-posedness of Non-Markovian Multidimensional Superquadratic BSDE," Papers 1912.03692, arXiv.org, revised Jan 2022.
    10. Hu, Ying & Lin, Yiqing & Soumana Hima, Abdoulaye, 2018. "Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3724-3750.
    11. Ying Hu & Gechun Liang & Shanjian Tang, 2018. "Systems of ergodic BSDEs arising in regime switching forward performance processes," Papers 1807.01816, arXiv.org, revised Jun 2020.
    12. Jackson, Joe & Žitković, Gordan, 2022. "A characterization of solutions of quadratic BSDEs and a new approach to existence," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 210-225.
    13. Jana Bielagk & Arnaud Lionnet & Gonçalo dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Working Papers hal-01245812, HAL.
    14. Vicky Henderson & Gechun Liang, 2014. "Pseudo linear pricing rule for utility indifference valuation," Finance and Stochastics, Springer, vol. 18(3), pages 593-615, July.
    15. Hao, Tao & Wen, Jiaqiang & Xiong, Jie, 2022. "Solvability of a class of mean-field BSDEs with quadratic growth," Statistics & Probability Letters, Elsevier, vol. 191(C).
    16. Martin Herdegen & Johannes Muhle-Karbe & Dylan Possamaï, 2021. "Equilibrium asset pricing with transaction costs," Finance and Stochastics, Springer, vol. 25(2), pages 231-275, April.
    17. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin's Differentiability," Papers 1512.05924, arXiv.org, revised Sep 2017.
    18. Fujii, Masaaki & Takahashi, Akihiko, 2018. "Quadratic–exponential growth BSDEs with jumps and their Malliavin’s differentiability," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 2083-2130.
    19. Geiss, Stefan & Ylinen, Juha, 2020. "Weighted bounded mean oscillation applied to backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3711-3752.
    20. Jana Bielagk & Arnaud Lionnet & Goncalo Dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Papers 1511.04218, arXiv.org, revised Feb 2017.

    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:141:y:2021:i:c:p:376-411. 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.