A class of stochastic optimal control problems with state constraint

Hltoshi Ishii, Paola Loreti

    Research output: Contribution to journalArticle

    13 Citations (Scopus)

    Abstract

    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.

    Original languageEnglish
    Pages (from-to)1167-1196
    Number of pages30
    JournalIndiana University Mathematics Journal
    Volume51
    Issue number5
    Publication statusPublished - 2002

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    Stochastic Optimal Control
    State Constraints
    Optimal Control Problem
    Viscosity Solutions
    Hölder Regularity
    Control Sets
    Hamilton-Jacobi-Bellman Equation
    Infinite Horizon
    Value Function
    Dynamic Programming
    Stochastic Equations
    Vanish
    Viscosity
    Class
    Differential equation
    Denote
    Subset

    Keywords

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

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

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

    In: Indiana University Mathematics Journal, Vol. 51, No. 5, 2002, p. 1167-1196.

    Research output: Contribution to journalArticle

    Ishii, Hltoshi ; Loreti, Paola. / A class of stochastic optimal control problems with state constraint. In: Indiana University Mathematics Journal. 2002 ; Vol. 51, No. 5. pp. 1167-1196.
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