Order

, Volume 22, Issue 2, pp 145–183 | Cite as

Regulating Functions on Partially Ordered Sets

Article
  • 42 Downloads

Abstract

We study the so-called Skorokhod reflection problem (SRP) posed for real-valued functions defined on a partially ordered set (poset), when there are two boundaries, considered also to be functions of the poset. The problem is to constrain the function between the boundaries by adding and subtracting nonnegative nondecreasing (NN) functions in the most efficient way. We show existence and uniqueness of its solution by using only order theoretic arguments. The solution is also shown to obey a fixed point equation. When the underlying poset is a σ-algebra of subsets of a set, our results yield a generalization of the classical Jordan–Hahn decomposition of a signed measure. We also study the problem on a poset that has the structure of a tree, where we identify additional structural properties of the solution, and on discrete posets, where we show that the fixed point equation uniquely characterizes the solution. Further interesting posets we consider are the poset of real n-vectors ordered by majorization, and the poset of n × n positive semidefinite real matrices ordered by pointwise ordering of the associated quadratic forms. We say a function on a poset is of bounded variation if it can be written as the difference of two NN functions. The solution to the SRP when the upper and lower boundaries are the identically zero function corresponds to the most efficient or minimal such representation of a function of bounded variation. Minimal representations for several important functions of bounded variation on several of the posets mentioned above are determined in this paper.

Key Words

Skorokhod reflection partially ordered sets monotonicity Jordan decomposition signed measures majorization positive semidefinite matrices bounded variation 

Mathematics Subject Classifications (2000)

Primary 06A06, 06B35, 26A45, 47H09, 47H10 Secondary 06B23, 15A57, 60K25, 60J65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, R. and Orey, S.: Small random perturbations of dynamical systems with reflecting boundary, Nagoya Math. J. 60 (1976), 189–216.MATHMathSciNetGoogle Scholar
  2. 2.
    Bernard, A. K. and El Kharroubi, A.: Régulation déterministes et stochastiques dans le premier ‘orthant’ de \(\mathbb{R}_{n} \), Stochastics and Stochastics Reports 34 (1991), 149–167.MathSciNetGoogle Scholar
  3. 3.
    Cépa, E.: Probléme de Skorokhod multivoque, The Annals of Probability 26 (1998), 500–532.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Dupuis, P. and Ishii, H.: On the Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics and Stochastics Reports 35 (1991), 31–62.MathSciNetGoogle Scholar
  5. 5.
    Dupuis, P. and Ramanan, K. Convex duality and the Skorokhod problem, parts I & II, Prob. Th. and Rel. Fields 115 (1999), 153–195; 197–236.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Harrison, J. M.: Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985.MATHGoogle Scholar
  7. 7.
    Harrison, J. M. and Reiman, M. I.: Reflected Brownian motion on an orthant, The Annals of Probability 9 (1981), 302–308.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hewitt, E. and Stromberg, K.: Real and Abstract Analysis, Springer, Berlin Heidelberg New York, 1955.Google Scholar
  9. 9.
    Horn, R. A. and Johnson, C. R.: Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985.MATHGoogle Scholar
  10. 10.
    Konstantopoulos, T.: The Skorokhod reflection problem for functions with discontinuities (contractive case), Preprint, Univ. of Texas at Austin. 1999 Reference [26] of [22].Google Scholar
  11. 11.
    Konstantopoulos, T. and Anantharam, V.: Optimal flow control schemes that regulate the burstiness of traffic, IEEE/ACM Trans. Networking 3 (1995), 423–432.CrossRefGoogle Scholar
  12. 12.
    Marshall, A. W. and Olkin, I.: Inequalities: Theory of Majorization and Applications, Mathematics in Science and Engineering, Vol. 143, Academic, New York, 1979.Google Scholar
  13. 13.
    Oksendal, B. K.: Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin Heidelberg New York, 1998.Google Scholar
  14. 14.
    Port, S. C, and Stone, C. J.: Brownian Motion and Classical Potential Theory, Academic, New York, 1978.MATHGoogle Scholar
  15. 15.
    Royden, H. L.: Real Analysis, Macmillan, New York, 1968.Google Scholar
  16. 16.
    Skorokhod, A. V.: Stochastic equations for diffusions in a bounded region, Theory of Probab. and its Applic. 6 (1961), 264–274.CrossRefGoogle Scholar
  17. 17.
    Stanley, R. P.: Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, Monterey, California, 1986.Google Scholar
  18. 18.
    Stroock, D. W. and Varadhan, S. R. S.: Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24 (1971), 147–225.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Stroock, D. W. and Varadhan, S. R. S.: Multidimensional Diffusion Processes, Springer, Berlin, 1979.MATHGoogle Scholar
  20. 20.
    Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), 163–177.MATHMathSciNetGoogle Scholar
  21. 21.
    Varadhan, S. R. S. and Williams, R. J.: Brownian morion in a wedge with oblique reflection, Comm. Pure Appl. Math. 37 (1985), 405–443.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Whitt, W.: The reflection map with discontinuities, Mathematics of Operations Research 26 (No. 3) (2001), 447–484 .CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Yosida, K.: Functional Analysis, Springer, Berlin Heidelberg New York, 1995.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  2. 2.School of Mathematical and Computer SciencesHeriot–Watt UniversityEdinburghUK

Personalised recommendations