### 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 |
---|---|

Pages (from-to) | 1167-1196 |

Number of pages | 30 |

Journal | Indiana University Mathematics Journal |

Volume | 51 |

Issue number | 5 |

Publication status | Published - 2002 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*51*(5), 1167-1196.

**A class of stochastic optimal control problems with state constraint.** / Ishii, Hltoshi; Loreti, Paola.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 51, no. 5, pp. 1167-1196.

}

TY - JOUR

T1 - A class of stochastic optimal control problems with state constraint

AU - Ishii, Hltoshi

AU - Loreti, Paola

PY - 2002

Y1 - 2002

N2 - We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state Xt is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that Xt e Ḡ for all t > 0 or by the condition that Xt ∈ G for all t > 0, where G be a given open subset of RN. Under the assumption that for each z ∈ ∂G there exists az ∈ A, where A denotes the control set, such that the diffusion matrix σ(x, a) vanishes for a = az and for x ∈ ∂G in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = az 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, Vr of the relaxed problem associated with the constraint Ḡ, and V0 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.

AB - We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state Xt is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that Xt e Ḡ for all t > 0 or by the condition that Xt ∈ G for all t > 0, where G be a given open subset of RN. Under the assumption that for each z ∈ ∂G there exists az ∈ A, where A denotes the control set, such that the diffusion matrix σ(x, a) vanishes for a = az and for x ∈ ∂G in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = az 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, Vr of the relaxed problem associated with the constraint Ḡ, and V0 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.

KW - Degenerate elliptic equations

KW - Hamilton-Jacobi-Bellman equations

KW - State constraint

KW - Stochastic optimal control

KW - Viscosity solutions

UR - http://www.scopus.com/inward/record.url?scp=0036997954&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036997954&partnerID=8YFLogxK

M3 - Article

VL - 51

SP - 1167

EP - 1196

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 5

ER -