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A stochastic approach to a multivalued Dirichlet-Neumann problem

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  • Maticiuc, Lucian
  • Rascanu, Aurel

Abstract

We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality (PVI) with a mixed nonlinear multivalued Neumann-Dirichlet boundary condition: where [not partial differential][phi] and [not partial differential][psi] are subdifferential operators and is a second-differential operator given by The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality: where (At)t>=0 is a continuous one-dimensional increasing measurable process, and then we obtain a Feynman-Kaç representation formula for the viscosity solution of the PVI problem.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:6:p:777-800
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    References listed on IDEAS

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    1. 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.
    2. Pardoux, Etienne & Rascanu, Aurel, 1998. "Backward stochastic differential equations with subdifferential operator and related variational inequalities," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 191-215, August.
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    1. 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.
    2. Cordoni, Francesco & Di Persio, Luca & Maticiuc, Lucian & Zălinescu, Adrian, 2020. "A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1669-1712.
    3. 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.
    4. Lu, Wen & Ren, Yong & Hu, Lanying, 2015. "Mean-field backward stochastic differential equations with subdifferential operator and its applications," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 73-81.

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