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Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations

Author

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  • N'Zi, Modeste
  • Ouknine, Youssef
  • Sulem, Agnès

Abstract

We study the regularity of the viscosity solution of a quasilinear parabolic partial differential equation with Lipschitz coefficients by using its connection with a forward backward stochastic differential equation (in short FBSDE) and we give a probabilistic representation of the generalized gradient (derivative in the distribution sense) of the viscosity solution. This representation is a kind of nonlinear Feynman-Kac formula. The main idea is to show that the FBSDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. If the diffusion coefficient of the forward equation is uniformly elliptic, we approximate the FBSDE by smooth ones and use Krylov's estimate to prove the convergence of the derivatives. In the degenerate case, we use techniques of Bouleau-Hirsch on absolute continuity of probability measures.

Suggested Citation

  • N'Zi, Modeste & Ouknine, Youssef & Sulem, Agnès, 2006. "Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1319-1339, September.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:9:p:1319-1339
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    Cited by:

    1. Richter, Anja, 2014. "Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3578-3611.
    2. Olivier Menoukeu-Pamen & Ludovic Tangpi, 2023. "Maximum Principle for Stochastic Control of SDEs with Measurable Drifts," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1195-1228, June.
    3. Jean-François Chassagneux & Romuald Elie & Idris Kharroubi, 2015. "When terminal facelift enforces delta constraints," Finance and Stochastics, Springer, vol. 19(2), pages 329-362, April.

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