Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

On the Solutions of the Problem for a Singular Ergodic Control

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper discusses an eigenvalue problem for a singular ergodic control. The eigenvalue has a probabilistic interpretation which can be regarded as the least, long-time averaged (ergodic) cost for a singular control problem. The existence and uniqueness of positive radial solutions of an equation with constraints involving gradient which is related to a stochastic optimal control problem under certain conditions on the nonlinearity of the equation are examined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Evans, L.C.: A second-order elliptic equation with gradient constraint. Commun. Partial Differ. Equ. 4(5), 555–572 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Evans, L.C.: Correction to: a second-order elliptic equation with gradient constraint. Commun. Partial Differ. Equ. 4(10), 1199 (1979)

    Google Scholar 

  3. Friedman, A.: Variational Principles and Free-Boundary Problems, 2nd edn. Robert E. Krieger Publishing Co. Inc., Malabar (1988)

    MATH  Google Scholar 

  4. Ishii, H., Koike, S.: Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Commun. Partial Differ. Equ. 8(4), 317–346 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Wiegner, M.: The \(C^{1-1}\)—character of solutions of second order elliptic equations with gradient constraint. Commun. Partial Differ. Equ. 6(3), 361–371 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hynd, R.: Analysis of Hamilton–Jacobi–Bellman equations arising in stochastic singular control. ESAIM Control Optim. Calc. Var. 19(1), 112–128 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Davis, M., Norman, R.: Portfolio selection with transaction costs. Math. Oper. Res. 15(4), 676–713 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davis, M., Panas, V., Zariphopoulou, T.: European option pricing under transaction costs. SIAM J. Cont. Opt. 31, 470–493 (1993)

    Article  MATH  Google Scholar 

  9. Hynd, R.: The eigenvalue problem of singular ergodic control. Commun. Pure Appl. Math. 65, 0649–0682 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Menaldi, J.-L., Robin, M., Taksar, M.: Singular ergodic control for multidimensional Gaussian processes. Math. Control Signals Syst. 5(1), 93–114 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Soner, H.M., Shreve, S.E.: Regularity of the value function for a two-dimensional singular stochastic control problem. SIAM J. Control Optim. 27(4), 876–907 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kruk, N.J.: Optimal policies for \(n\)-dimensional singular stochastic control problems. Part II: the radially symmetric case. Ergodic control. SIAM J. Control Optim. 39(2), 635–659 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees and the editor for their valuable comments which have helped improve the presentation of this paper. This work is partially supported by the Ministry of Science and Technology of Taiwan under Grant No. MOST 103-2115-M-018-002-MY3.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Central University, Taoyuan, Taiwan

    Yen-Lin Wu

  2. Department of Mathematics, National Changhua University of Education, Changhua, Taiwan

    Zhi-You Chen

Authors
  1. Yen-Lin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-You Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-You Chen.

Additional information

Mark J. Balas.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, YL., Chen, ZY. On the Solutions of the Problem for a Singular Ergodic Control. J Optim Theory Appl 173, 746–762 (2017). https://doi.org/10.1007/s10957-017-1099-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1099-y

Keywords

Mathematics Subject Classification

Search

Navigation