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Reflected forward-backward SDE s and obstacle problems with boundary conditions

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  • Jin Ma
  • Jakša Cvitanić

Abstract

In this paper we study a class of forward-backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may depend on time and is possibly random. The solvability of such FBSDER is studied in a fairly general way. We also prove that if the coefficients are all deterministic and the backward equation is one-dimensional, then the adapted solution of such FBSDER will give the viscosity solution of a quasilinear variational inequality (obstacle problem) with a Neumann boundary condition. As an application, we study how the solvability of FBSDERs is related to the solvability of an American game option .

Suggested Citation

  • Jin Ma & Jakša Cvitanić, 2001. "Reflected forward-backward SDE s and obstacle problems with boundary conditions," International Journal of Stochastic Analysis, Hindawi, vol. 14, pages 1-26, January.
    • Handle: RePEc:hin:jnijsa:402830
    DOI: 10.1155/S1048953301000090
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    Cited by:

    1. Ning, Ning & Wu, Jing & Zheng, Jinwei, 2024. "One-dimensional McKean–Vlasov stochastic variational inequalities and coupled BSDEs with locally Hölder noise coefficients," Stochastic Processes and their Applications, Elsevier, vol. 171(C).
    2. Qikang Ran & Tusheng Zhang, 2010. "Existence and Uniqueness of Bounded Weak Solutions of a Semilinear Parabolic PDE," Journal of Theoretical Probability, Springer, vol. 23(4), pages 951-971, December.
    3. Zhen-Qing Chen & Xinwei Feng, 2021. "Reflected Backward Stochastic Differential Equation with Rank-Based Data," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1213-1247, September.

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