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Switching problem and related system of reflected backward SDEs

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  • Hamadène, Said
  • Zhang, Jianfeng

Abstract

This paper studies a system of backward stochastic differential equations with oblique reflections (RBSDEs for short), motivated by the switching problem under Knightian uncertainty and recursive utilities. The main feature of our system is that its components are interconnected through both the generators and the obstacles. We prove existence, uniqueness, and stability of the solution of the RBSDE, and give the expression of the price and the optimal strategy for the original switching problem via a verification theorem.

Suggested Citation

  • Hamadène, Said & Zhang, Jianfeng, 2010. "Switching problem and related system of reflected backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 403-426, April.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:4:p:403-426
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    1. Dixit, Avinash K, 1989. "Entry and Exit Decisions under Uncertainty," Journal of Political Economy, University of Chicago Press, vol. 97(3), pages 620-638, June.
    2. Lenos Trigeorgis, 1993. "Real Options and Interactions With Financial Flexibility," Financial Management, Financial Management Association, vol. 22(3), Fall.
    3. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    4. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    5. Said Hamadène & Monique Jeanblanc, 2007. "On the Starting and Stopping Problem: Application in Reversible Investments," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 182-192, February.
    6. Hamadène, S. & Lepeltier, J. -P., 2000. "Reflected BSDEs and mixed game problem," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 177-188, February.
    7. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    8. Duffie, Darrel & Lions, Pierre-Louis, 1992. "PDE solutions of stochastic differential utility," Journal of Mathematical Economics, Elsevier, vol. 21(6), pages 577-606.
    9. Brennan, Michael J & Schwartz, Eduardo S, 1985. "Evaluating Natural Resource Investments," The Journal of Business, University of Chicago Press, vol. 58(2), pages 135-157, April.
    10. Boualem Djehiche & Said Hamadène, 2009. "On A Finite Horizon Starting And Stopping Problem With Risk Of Abandonment," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(04), pages 523-543.
    11. Guo, Xin & Pham, Huyên, 2005. "Optimal partially reversible investment with entry decision and general production function," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 705-736, May.
    12. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
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    Cited by:

    1. Aazizi, Soufiane & El Mellali, Tarik & Fakhouri, Imade & Ouknine, Youssef, 2018. "Optimal switching problem and related system of BSDEs with left-Lipschitz coefficients and mixed reflections," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 70-78.
    2. Eddahbi, M’hamed & Fakhouri, Imade & Ouknine, Youssef, 2020. "Reflected BSDEs with jumps in time-dependent convex càdlàg domains," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6515-6555.
    3. Matoussi, Anis & Sabbagh, Wissal & Zhang, Tusheng, 2017. "Backward doubly SDEs and semilinear stochastic PDEs in a convex domain," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2781-2815.
    4. Chassagneux, Jean-François & Richou, Adrien, 2019. "Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4597-4637.
    5. Erhan Bayraktar & Qi Feng & Zhaoyu Zhang, 2022. "Deep Signature Algorithm for Multi-dimensional Path-Dependent Options," Papers 2211.11691, arXiv.org, revised Jan 2024.
    6. Mihail Zervos & Carlos Oliveira & Kate Duckworth, 2018. "An investment model with switching costs and the option to abandon," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 417-443, December.
    7. El Asri, Brahim, 2013. "Stochastic optimal multi-modes switching with a viscosity solution approach," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 579-602.
    8. Yuki Shigeta, 2016. "Optimal Switching under Ambiguity and Its Applications in Finance," Papers 1608.06045, arXiv.org.
    9. Chi Seng Pun, 2022. "Robust classical-impulse stochastic control problems in an infinite horizon," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 291-312, October.
    10. Klimsiak, Tomasz, 2019. "Systems of quasi-variational inequalities related to the switching problem," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1259-1286.
    11. Li Kai & Nyström Kaj & Olofsson Marcus, 2015. "Optimal switching problems under partial information," Monte Carlo Methods and Applications, De Gruyter, vol. 21(2), pages 91-120, June.
    12. Fuhrman, Marco & Morlais, Marie-Amélie, 2020. "Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3120-3153.
    13. Bénézet, Cyril & Chassagneux, Jean-François & Richou, Adrien, 2022. "Switching problems with controlled randomisation and associated obliquely reflected BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 23-71.
    14. Kaitong Hu & Zhenjie Ren & Junjian Yang, 2019. "Principal-agent problem with multiple principals," Working Papers hal-02088486, HAL.
    15. Cacace, S. & Ferretti, R. & Festa, A., 2020. "Stochastic hybrid differential games and match race problems," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    16. Giovanni Mottola, 2014. "A stochastic switching control model arising in general OTC contracts with contingent CSA in presence of CVA, collateral and funding," Papers 1412.1469, arXiv.org.
    17. Elie, Romuald & Kharroubi, Idris, 2010. "Probabilistic representation and approximation for coupled systems of variational inequalities," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1388-1396, September.
    18. Magnus Perninge, 2020. "A finite horizon optimal switching problem with memory and application to controlled SDDEs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(3), pages 465-500, June.
    19. Nie, Tianyang & Rutkowski, Marek, 2014. "Multi-player stopping games with redistribution of payoffs and BSDEs with oblique reflection," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2672-2698.

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