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

A Probabilistic Approach to Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 67 (3) – Jun 1, 2013
34 pages

/lp/springer_journal/a-probabilistic-approach-to-interior-regularity-of-fully-nonlinear-E2iNxwYFQ6
Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9194-4
Publisher site
See Article on Publisher Site

### 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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2013

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations