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A Probabilistic Approach to Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations in Smooth Domains

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Abstract

We consider the value function of a stochastic optimal control of degenerate diffusion processes in a domain D. We study the smoothness of the value function, under the assumption of the non-degeneracy of the diffusion term along the normal to the boundary and an interior condition weaker than the non-degeneracy of the diffusion term. When the diffusion term, drift term, discount factor, running payoff and terminal payoff are all in the class of \(C^{1,1}(\bar{D})\), the value function turns out to be the unique solution in the class of \(C_{loc}^{1,1}(D)\cap C^{0,1}(\bar{D})\) to the associated degenerate Bellman equation with Dirichlet boundary data. Our approach is probabilistic.

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Acknowledgement

The author is sincerely grateful to his advisor, N.V. Krylov, for giving many useful suggestions on the improvements. The author also would like to thank the referee for pointing out several misprints and mistakes and giving comments on the manuscript of this article.

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  1. School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN, 55455, USA

    Wei Zhou

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  1. Wei Zhou
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Correspondence to Wei Zhou.

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Zhou, W. A Probabilistic Approach to Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations in Smooth Domains. Appl Math Optim 67, 419–452 (2013). https://doi.org/10.1007/s00245-013-9194-4

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