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Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes

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  • Dirk Becherer
  • Martin Schweizer

Abstract

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuation problems for contingent claims with a recursive payoff structure.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:math/0505208
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    References listed on IDEAS

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    1. Thomas Møller, 2001. "Risk-minimizing hedging strategies for insurance payment processes," Finance and Stochastics, Springer, vol. 5(4), pages 419-446.
    2. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    3. Christophette Blanchet-Scalliet & Monique Jeanblanc, 2004. "Hazard rate for credit risk and hedging defaultable contingent claims," Finance and Stochastics, Springer, vol. 8(1), pages 145-159, January.
    4. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
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    Cited by:

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    2. Agostino Capponi & Christoph Frei, 2017. "Systemic Influences on Optimal Equity-Credit Investment," Management Science, INFORMS, vol. 63(8), pages 2756-2771, August.
    3. 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.
    4. Kim, Jinbeom & Leung, Tim, 2016. "Pricing derivatives with counterparty risk and collateralization: A fixed point approach," European Journal of Operational Research, Elsevier, vol. 249(2), pages 525-539.
    5. Akira Kashiwabara & Nobuhiro Nakamura, 2011. "Dynamic Investment Strategies to Reaction–Diffusion Systems Based upon Stochastic Differential Utilities," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 18(2), pages 131-150, May.
    6. John R. Birge & Lijun Bo & Agostino Capponi, 2018. "Risk-Sensitive Asset Management and Cascading Defaults," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 1-28, February.
    7. 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.
    8. 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.
    9. Lijun Bo & Agostino Capponi, 2018. "Portfolio Choice with Market-Credit Risk Dependencies," Papers 1806.07175, arXiv.org.
    10. Johannes Muhle-Karbe & Marcel Nutz & Xiaowei Tan, 2019. "Asset Pricing with Heterogeneous Beliefs and Illiquidity," Papers 1905.05730, arXiv.org, revised Mar 2020.
    11. Tim Leung & Ronnie Sircar, 2009. "Accounting For Risk Aversion, Vesting, Job Termination Risk And Multiple Exercises In Valuation Of Employee Stock Options," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 99-128, January.
    12. Álvaro Cartea & Sebastian Jaimungal & Damir Kinzebulatov, 2016. "Algorithmic Trading With Learning," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-30, June.
    13. Giorgia Callegaro & Luciano Campi & Valeria Giusto & Tiziano Vargiolu, 2017. "Utility indifference pricing and hedging for structured contracts in energy markets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(2), pages 265-303, April.
    14. Kraft, Holger & Steffensen, Mogens, 2009. "Asset allocation with contagion and explicit bankruptcy procedures," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 147-167, January.
    15. Damiano Brigo & Federico Graceffa & Alexander Kalinin, 2021. "Mild to classical solutions for XVA equations under stochastic volatility," Papers 2112.11808, arXiv.org.
    16. Stéphane Crépey & Shiqi Song, 2014. "Counterparty risk and funding: Immersion and beyond," Working Papers hal-00989062, HAL.
    17. Łukasz Delong & Russell Gerrard, 2007. "Mean-variance portfolio selection for a non-life insurance company," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(2), pages 339-367, October.
    18. Liang, Zongxia & Zhao, Xiaoyang, 2016. "Optimal mean–variance efficiency of a family with life insurance under inflation risk," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 164-178.
    19. Jacek Jakubowski & Mariusz Niewk{e}g{l}owski, 2013. "Risk-minimization and hedging claims on a jump-diffusion market model, Feynman-Kac Theorem and PIDE," Papers 1305.4132, arXiv.org, revised Jul 2013.
    20. Johannes Muhle‐Karbe & Marcel Nutz & Xiaowei Tan, 2020. "Asset pricing with heterogeneous beliefs and illiquidity," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1392-1421, October.

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