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On the continuity of the probabilistic representation of a semilinear Neumann–Dirichlet problem

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  • Maticiuc, Lucian
  • Răşcanu, Aurel

Abstract

In this article we prove the continuity of the deterministic function u:[0,T]×D̄→R, defined by u(t,x):=Ytt,x, where the process (Yst,x)s∈[t,T] is given by the generalized multivalued backward stochastic differential equation:{−dYst,x+∂φ(Yst,x)ds+∂ψ(Yst,x)dAst,x∋f(s,Xst,x,Yst,x)ds+g(s,Xst,x,Yst,x)dAst,x−Zst,xdWs,t≤s

Suggested Citation

  • Maticiuc, Lucian & Răşcanu, Aurel, 2016. "On the continuity of the probabilistic representation of a semilinear Neumann–Dirichlet problem," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 572-607.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:2:p:572-607
    DOI: 10.1016/j.spa.2015.09.011
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    References listed on IDEAS

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    1. Maticiuc, Lucian & Rascanu, Aurel, 2010. "A stochastic approach to a multivalued Dirichlet-Neumann problem," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 777-800, June.
    2. Aman, Auguste & Mrhardy, Naoul, 2013. "Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 863-874.
    3. Richou, Adrien, 2009. "Ergodic BSDEs and related PDEs with Neumann boundary conditions," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2945-2969, September.
    4. Boufoussi, B. & van Casteren, J., 2004. "An approximation result for a nonlinear Neumann boundary value problem via BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 331-350, December.
    5. Hu, Ying, 1993. "Probabilistic interpretation of a system of quasilinear elliptic partial differential equations under Neumann boundary conditions," Stochastic Processes and their Applications, Elsevier, vol. 48(1), pages 107-121, October.
    6. Lejay, Antoine, 2002. "BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 1-39, January.
    7. Gassous, Anouar M. & Răşcanu, Aurel & Rotenstein, Eduard, 2015. "Multivalued backward stochastic differential equations with oblique subgradients," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3170-3195.
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