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Multivalued backward stochastic differential equations with oblique subgradients

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  • Gassous, Anouar M.
  • Răşcanu, Aurel
  • Rotenstein, Eduard

Abstract

We study the existence and uniqueness of the solution for the following backward stochastic variational inequality with oblique reflection (for short, BSVI(H(t,y)∂φ(y))), written under differential form {−dYt+H(t,Yt)∂φ(Yt)(dt)∋F(t,Yt,Zt)dt−ZtdBt,t∈[0,T],YT=η, where H is a bounded symmetric smooth matrix and φ is a proper convex lower semicontinuous function, with ∂φ being its subdifferential operator. The presence of the product H∂φ does not permit the use of standard techniques because it conserves neither the Lipschitz property of the matrix nor the monotonicity property of the subdifferential operator. We prove that, if we consider the dependence of H only on the time, the equation admits a unique strong solution and, allowing the dependence on the state of the system, the above BSVI(H(t,y)∂φ(y)) admits a weak solution in the sense of the Meyer–Zheng topology. However, for that purpose we must renounce at the dependence on Z for the generator function and we situate our problem in a Markovian framework.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:8:p:3170-3195
    DOI: 10.1016/j.spa.2015.03.001
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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. Gassous, Anouar M. & Răşcanu, Aurel & Rotenstein, Eduard, 2012. "Stochastic variational inequalities with oblique subgradients," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2668-2700.
    3. 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.
    4. 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.
    5. 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.
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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.

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