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Stochastic partial differential equations with singular terminal condition

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  • Matoussi, A.
  • Piozin, L.
  • Popier, A.

Abstract

In this paper, we first prove existence and uniqueness of the solution of a backward doubly stochastic differential equation (BDSDE) and of the related stochastic partial differential equation (SPDE) under monotonicity assumption on the generator. Then we study the case where the terminal data is singular, in the sense that it can be equal to +∞ on a set with positive measure. In this setting we show that there exists a minimal solution, both for the BDSDE and for the SPDE. Note that solution of the SPDE means weak solution in the Sobolev sense.

Suggested Citation

  • Matoussi, A. & Piozin, L. & Popier, A., 2017. "Stochastic partial differential equations with singular terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 831-876.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:3:p:831-876
    DOI: 10.1016/j.spa.2016.07.002
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    References listed on IDEAS

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    1. Popier, A., 2006. "Backward stochastic differential equations with singular terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 2014-2056, December.
    2. 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.
    3. Buckdahn, Rainer & Ma, Jin, 2001. "Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 205-228, June.
    4. Buckdahn, Rainer & Ma, Jin, 2001. "Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 181-204, June.
    5. Philippe Briand & René Carmona, 2000. "BSDEs with polynomial growth generators," International Journal of Stochastic Analysis, Hindawi, vol. 13, pages 1-32, January.
    6. Stoica, I. L., 2003. "A probabilistic interpretation of the divergence and BSDE's," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 31-55, January.
    7. Briand, Ph. & Delyon, B. & Hu, Y. & Pardoux, E. & Stoica, L., 2003. "Lp solutions of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 109-129, November.
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