IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1607.06644.html
   My bibliography  Save this paper

On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

Author

Listed:
  • Dirk Becherer
  • Martin Buttner
  • Klebert Kentia

Abstract

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.

Suggested Citation

  • Dirk Becherer & Martin Buttner & Klebert Kentia, 2016. "On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples," Papers 1607.06644, arXiv.org, revised Nov 2019.
    • Handle: RePEc:arx:papers:1607.06644
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1607.06644
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. John H. Cochrane & Jesus Saa-Requejo, 2000. "Beyond Arbitrage: Good-Deal Asset Price Bounds in Incomplete Markets," Journal of Political Economy, University of Chicago Press, vol. 108(1), pages 79-119, February.
    2. Marcel Nutz, 2009. "The Opportunity Process for Optimal Consumption and Investment with Power Utility," Papers 0912.1879, arXiv.org, revised Jun 2010.
    3. Dirk Becherer & Martin Schweizer, 2005. "Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes," Papers math/0505208, arXiv.org.
    4. Roger J. A. Laeven & Mitja Stadje, 2014. "Robust Portfolio Choice and Indifference Valuation," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1109-1141, November.
    5. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    6. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    7. Geiss, Christel & Labart, Céline, 2016. "Simulation of BSDEs with jumps by Wiener Chaos expansion," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2123-2162.
    8. Marcel Nutz, 2009. "The Bellman equation for power utility maximization with semimartingales," Papers 0912.1883, arXiv.org, revised Mar 2012.
    9. 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.
    10. Confortola, Fulvia & Fuhrman, Marco, 2014. "Backward stochastic differential equations associated to jump Markov processes and applications," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 289-316.
    11. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
    12. Samuel N. Cohen & Robert J. Elliott, 2008. "Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions," Papers 0810.0055, arXiv.org, revised Jan 2010.
    13. Quenez, Marie-Claire & Sulem, Agnès, 2013. "BSDEs with jumps, optimization and applications to dynamic risk measures," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3328-3357.
    14. 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.
    15. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
    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. 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.
    2. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin's Differentiability," Papers 1512.05924, arXiv.org, revised Sep 2017.
    3. Madan, Dilip & Pistorius, Martijn & Stadje, Mitja, 2016. "Convergence of BSΔEs driven by random walks to BSDEs: The case of (in)finite activity jumps with general driver," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1553-1584.
    4. Masaaki Fujii & Akihiko Takahashi, 2016. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin’s Differentiability (revised version of CARF-F-376)," CARF F-Series CARF-F-395, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Thai Nguyen & Mitja Stadje, 2020. "Utility maximization under endogenous pricing," Papers 2005.04312, arXiv.org, revised Mar 2024.
    6. 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.
    7. M. Nabil Kazi-Tani & Dylan Possamai & Chao Zhou, 2014. "Quadratic BSDEs with jumps: related non-linear expectations," Papers 1403.2730, arXiv.org.
    8. Richter, Anja, 2014. "Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3578-3611.
    9. 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.
    10. Masaaki Fujii & Akihiko Takahashi, 2017. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin's Differentiability (Forthcoming in Stochastic Processes and their Applications)(revised version of CARF-F-395)," CARF F-Series CARF-F-420, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    11. Antoon Pelsser & Mitja Stadje, 2014. "Time-Consistent And Market-Consistent Evaluations," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 25-65, January.
    12. Ying Hu & Xiaomin Shi & Zuo Quan Xu, 2022. "Optimal consumption-investment with coupled constraints on consumption and investment strategies in a regime switching market with random coefficients," Papers 2211.05291, arXiv.org.
    13. Roger J. A. Laeven & Mitja Stadje, 2014. "Robust Portfolio Choice and Indifference Valuation," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1109-1141, November.
    14. Antonelli, Fabio & Mancini, Carlo, 2016. "Solutions of BSDE’s with jumps and quadratic/locally Lipschitz generator," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 3124-3144.
    15. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential growth BSDEs with Jumps and their Malliavin’s Differentiability," CARF F-Series CARF-F-376, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    16. Łukasz Delong, 2019. "Optimal investment for insurance company with exponential utility and wealth-dependent risk aversion coefficient," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(1), pages 73-113, February.
    17. Frei, Christoph & Mocha, Markus & Westray, Nicholas, 2012. "BSDEs in utility maximization with BMO market price of risk," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2486-2519.
    18. Hiroaki Hata & Shuenn-Jyi Sheu & Li-Hsien Sun, 2019. "Expected exponential utility maximization of insurers with a general diffusion factor model : The complete market case," Papers 1903.08957, arXiv.org.
    19. Masaaki Fujii & Akihiko Takahashi, 2015. "Quadratic-exponential Growth BSDEs with Jumps and their Malliavin's Differentiability," CIRJE F-Series CIRJE-F-997, CIRJE, Faculty of Economics, University of Tokyo.
    20. Balter, Anne G. & Pelsser, Antoon, 2020. "Pricing and hedging in incomplete markets with model uncertainty," European Journal of Operational Research, Elsevier, vol. 282(3), pages 911-925.

    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:1607.06644. 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.