IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1510.03220.html
   My bibliography  Save this paper

Asymptotic Expansion for Forward-Backward SDEs with Jumps

Author

Listed:
  • Masaaki Fujii
  • Akihiko Takahashi

Abstract

This work provides a semi-analytic approximation method for decoupled forwardbackward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with {\sigma}-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.

Suggested Citation

  • Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," Papers 1510.03220, arXiv.org, revised Sep 2018.
    • Handle: RePEc:arx:papers:1510.03220
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1510.03220
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Akihiko Takahashi & Toshihiro Yamada, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CARF F-Series CARF-F-363, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. Masaaki Fujii, Akihiko Takahashi, 2012. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," CARF F-Series CARF-F-269, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Masaaki Fujii, 2015. "Optimal Position Management for a Market Maker with Stochastic Price Impacts," CIRJE F-Series CIRJE-F-963, CIRJE, Faculty of Economics, University of Tokyo.
    4. Bouchard, Bruno & Elie, Romuald, 2008. "Discrete-time approximation of decoupled Forward-Backward SDE with jumps," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 53-75, January.
    5. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    6. M. Mania & R. Tevzadze, 2003. "Backward Stochastic PDE and Imperfect Hedging," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 6(07), pages 663-692.
    7. Masaaki Fujii, 2015. "Optimal Position Management for a Market Maker with Stochastic Price Impacts," Papers 1503.07007, arXiv.org, revised Sep 2015.
    8. Hu, Ying & Yong, Jiongmin, 2000. "Forward-backward stochastic differential equations with nonsmooth coefficients," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 93-106, May.
    9. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CARF F-Series CARF-F-278, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2015.
    10. Akihiko Takahashi, 2015. "Asymptotic Expansion Approach in Finance," CARF F-Series CARF-F-356, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    11. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    12. 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.
    13. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs," CARF F-Series CARF-F-372, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    14. Masaaki Fujii & Akihiko Takahashi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 283-304, September.
    15. Masaaki Fujii, 2015. "Optimal Position Management for a Market Maker with Stochastic Price Impacts," CARF F-Series CARF-F-360, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2015.
    16. Delong, Lukasz & Imkeller, Peter, 2010. "On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1748-1775, August.
    17. Masaaki Fujii & Akihiko Takahshi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CIRJE F-Series CIRJE-F-954, CIRJE, Faculty of Economics, University of Tokyo.
    18. Masaaki Fujii & Akihiko Takahashi, 2011. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," Papers 1106.0123, arXiv.org, revised Jan 2012.
    19. Akihiko Takahashi & Toshihiro Yamada, 2015. "An asymptotic expansion of forward-backward SDEs with a perturbed driver," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-29.
    20. Pham, Huyên, 2010. "Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1795-1820, August.
    21. Masaaki Fujii & Akihiko Takahashi, 2012. "ANALYTICAL APPROXIMATION FOR NON-LINEAR FBSDEs WITH PERTURBATION SCHEME," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(05), pages 1-24.
    22. Akihiko Takahashi & Toshiaki Watanabe, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver ," CIRJE F-Series CIRJE-F-976, CIRJE, Faculty of Economics, University of Tokyo.
    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. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," Papers 1710.07030, arXiv.org, revised Mar 2019.
    2. 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.
    3. Stéphane Crépey & Shiqi Song, 2018. "Counterparty risk and funding: immersion and beyond," Working Papers hal-01764403, HAL.
    4. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions," CIRJE F-Series CIRJE-F-1016, CIRJE, Faculty of Economics, University of Tokyo.
    5. Stéphane Crépey & Shiqi Song, 2016. "Counterparty risk and funding: immersion and beyond," Finance and Stochastics, Springer, vol. 20(4), pages 901-930, October.
    6. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CIRJE F-Series CIRJE-F-1069, CIRJE, Faculty of Economics, University of Tokyo.
    7. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CARF F-Series CARF-F-423, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.

    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. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," CIRJE F-Series CIRJE-F-993, CIRJE, Faculty of Economics, University of Tokyo.
    2. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs," CARF F-Series CARF-F-372, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Masaaki Fujii & Akihiko Takahashi, 2018. "Asymptotic Expansion for Forward-Backward SDEs with JumpsAsymptotic Expansion for Forward-Backward SDEs with Jumps (Forthcoming in Stochastics) (Revised version of F-372)," CARF F-Series CARF-F-445, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    4. Fujii, Masaaki & Takahashi, Akihiko, 2019. "Solving backward stochastic differential equations with quadratic-growth drivers by connecting the short-term expansions," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1492-1532.
    5. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations with quadratic-growth drivers by Connecting the Short-term Expansions," Papers 1606.04285, arXiv.org, revised May 2018.
    6. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 391-408, September.
    7. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions(Revised version of CARF-F-387)," CARF F-Series CARF-F-398, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    8. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions," CIRJE F-Series CIRJE-F-1016, CIRJE, Faculty of Economics, University of Tokyo.
    9. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions," CARF F-Series CARF-F-387, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    10. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," Papers 1710.07030, arXiv.org, revised Mar 2019.
    11. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CIRJE F-Series CIRJE-F-1069, CIRJE, Faculty of Economics, University of Tokyo.
    12. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs (Forthcoming in Asia-Pacific Financial Markets)," CARF F-Series CARF-F-456, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    13. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CARF F-Series CARF-F-423, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    14. Stéphane Crépey & Shiqi Song, 2016. "Counterparty risk and funding: immersion and beyond," Finance and Stochastics, Springer, vol. 20(4), pages 901-930, October.
    15. Akihiko Takahashi & Toshihiro Yamada, 2016. "An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 23(4), pages 337-373, December.
    16. Masaaki Fujii & Akihiko Takahashi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 283-304, September.
    17. 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.
    18. Stéphane Crépey & Shiqi Song, 2018. "Counterparty risk and funding: immersion and beyond," Working Papers hal-01764403, HAL.
    19. Akihiko Takahashi & Yoshifumi Tsuchida & Toshihiro Yamada, 2021. "A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver," Papers 2101.09890, arXiv.org, revised Jan 2021.
    20. Masaaki Fujii & Akihiko Takahshi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CIRJE F-Series CIRJE-F-954, CIRJE, Faculty of Economics, University of Tokyo.

    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:1510.03220. 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.