### Abstract

We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state X_{t} is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that X_{t} e Ḡ for all t > 0 or by the condition that X_{t} ∈ G for all t > 0, where G be a given open subset of R^{N}. Under the assumption that for each z ∈ ∂G there exists a_{z} ∈ A, where A denotes the control set, such that the diffusion matrix σ(x, a) vanishes for a = a_{z} and for x ∈ ∂G in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = a_{z} and x = z as well as some other mild assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-Jacobi-Bellman equation, prove that the value functions V associated with the constraint Ḡ, V_{r} of the relaxed problem associated with the constraint Ḡ, and V_{0} associated with the constraint G, satisfy in the viscosity sense the state constraint problem, and establish Holder regularity results for the viscosity solution of the state constraint problem.

Original language | English |
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Pages (from-to) | 1167-1196 |

Number of pages | 30 |

Journal | Indiana University Mathematics Journal |

Volume | 51 |

Issue number | 5 |

Publication status | Published - 2002 |

### Keywords

- Degenerate elliptic equations
- Hamilton-Jacobi-Bellman equations
- State constraint
- Stochastic optimal control
- Viscosity solutions

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Indiana University Mathematics Journal*,

*51*(5), 1167-1196.