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Adapted solution of a degenerate backward spde, with applications

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  • Ma, Jin
  • Yong, Jiongmin

Abstract

In this paper we prove the existence and uniqueness, as well as the regularity, of the adapted solution to a class of degenerate linear backward stochastic partial differential equations (BSPDE) of parabolic type. We apply the results to a class of forward-backward stochastic differential equations (FBSDE) with random coefficients, and establish in a special case some explicit formulas among the solutions of FBSDEs and BSPDEs, including those involving Malliavin calculus. These relations lead to an adapted version of stochastic Feynman-Kac formula, as well as a stochastic Black-Scholes formula in mathematical finance.

Suggested Citation

  • Ma, Jin & Yong, Jiongmin, 1997. "Adapted solution of a degenerate backward spde, with applications," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 59-84, October.
    • Handle: RePEc:eee:spapps:v:70:y:1997:i:1:p:59-84
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    Citations

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    Cited by:

    1. Du, Kai & Zhang, Qi, 2013. "Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1616-1637.
    2. Englezos, Nikolaos & Frangos, Nikolaos E. & Kartala, Xanthi-Isidora & Yannacopoulos, Athanasios N., 2013. "Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3239-3272.
    3. McDonald, Stuart, 2006. "Finite Difference Approximation for Linear Stochastic Partial Differential Equations with Method of Lines," MPRA Paper 3983, University Library of Munich, Germany, revised 30 May 2007.
    4. Ma, Jin & Yin, Hong & Zhang, Jianfeng, 2012. "On non-Markovian forward–backward SDEs and backward stochastic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 3980-4004.
    5. Qiu, Jinniao, 2017. "Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1926-1959.
    6. Du, Kai & Meng, Qingxin, 2010. "A revisit to -theory of super-parabolic backward stochastic partial differential equations in," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1996-2015, September.
    7. Yin, Hong, 2014. "Solvability of forward–backward stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2583-2604.
    8. Lüders, Erik & Peisl, Bernhard, 2001. "How do investors' expectations drive asset prices?," ZEW Discussion Papers 01-15, ZEW - Leibniz Centre for European Economic Research.
    9. Confortola, Fulvia, 2007. "Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 613-628, May.
    10. Xanthi-Isidora Kartala & Nikolaos Englezos & Athanasios N. Yannacopoulos, 2020. "Future Expectations Modeling, Random Coefficient Forward–Backward Stochastic Differential Equations, and Stochastic Viscosity Solutions," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 403-433, May.
    11. Sundar, P. & Yin, Hong, 2009. "Existence and uniqueness of solutions to the backward 2D stochastic Navier-Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1216-1234, April.
    12. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.

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