Probability Theory and Related Fields

, Volume 96, Issue 3, pp 283–317 | Cite as

Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant

  • L. M. Taylor
  • R. J. Williams
Article

Summary

This work is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in the non-negative orthant of ℝd. Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the (d-1)-dimensional faces that form the boundary of the orthant, the bounded variation part of the process increases in a given direction (constant for any particular face) so as to confine the process to the orthant. For historical reasons, this “pushing” at the boundary is called instantaneous reflection. In 1988, Reiman and Williams proved that a necessary condition for the existence of such a semimartingale reflecting Brownian motion (SRBM) is that the reflection matrix formed by the directions of reflection be completely-L. In this work we prove that condition is sufficient for the existence of an SRBM and that the SRBM is unique in law. It follows from the uniqueness that an SRBM defines a strong Markov process. Our results have potential application to the study of diffusions arising as approximations tomulti-class queueing networks.

Mathematics Subject Classification (1991)

60J60 60J65 60G44 60K25 58G32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bass, R.F., Pardoux, E.: Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Relat. Fields76, 557–572 (1987)Google Scholar
  2. 2.
    Bernard, A., El Kharroubi, A.: Régulation de processus dans le premier orthant de ℝn. Stochastics Stochastics Rep.34, 149–167 (1991)Google Scholar
  3. 3.
    Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968Google Scholar
  4. 4.
    Cottle, R.W.: Completely-Q matrices. Math. Program.19, 347–351 (1980)Google Scholar
  5. 5.
    Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Ration. Mech. Anal.65, 275–288 (1977)Google Scholar
  6. 6.
    Dai, J.G., Harrison, J.M.: Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann. Appl. Probab.2, 65–86 (1992)Google Scholar
  7. 7.
    Dai, J.G., Kurtz, T.G.: The sufficiency of the basic adjoint relationship (in preparation)Google Scholar
  8. 8.
    Dupuis, P., Ishii, H.: On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics35, 31–62 (1991)Google Scholar
  9. 9.
    El Karoui, N., Chaleyat-Maurel, M.: Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur ℝ, Cas continu, in Temps Locaux. Astérisque52–53, 117–144 (1978)Google Scholar
  10. 10.
    Ethier, S.N., Kurtz, T.G.: Markov processes: characterization and convergence. New York: Wiley 1986Google Scholar
  11. 11.
    Hall, P., Heyde, C.C.: Martingale limit theory and its application. New York: Academic Press 1980Google Scholar
  12. 12.
    Harrison, J.M.: Brownian models of queueing networks with heterogeneous customer populations. In: Fleming, W., Lions, P.L. (eds.) Stochastic differential systems, stochastic control and applications. (IMA, vol. 10, pp. 147–186) Berlin Heidelberg New York: Springer 1988Google Scholar
  13. 13.
    Harrison, J.M., Reiman, M.I.: Reflected Brownian motion on an orthant. Ann. Probab.9, 302–308 (1981)Google Scholar
  14. 14.
    Harrison, J.M., Nguyen, V.: The QNET method for two-moment analysis of open queueing networks. Queueing Syst.6, 1–32 (1990)Google Scholar
  15. 15.
    Harrison, J.M., Nguyen, V.: Brownian models of multiclass queueing networks: current status and open problems. Queueing Syst. (to appear)Google Scholar
  16. 16.
    Harrison, J.M., Williams, R.J.: Brownian models of multiclass queueing networks. In: Proc. 29th I.E.E.E. Conf. on Decision and Control, 1990Google Scholar
  17. 17.
    Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Trans. Am. Math. Soc., I. Ser.10, 199–325 (1962)Google Scholar
  18. 18.
    Kurtz, T.G., Protter, Ph.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab.19, 1035–1070 (1991)Google Scholar
  19. 19.
    Kwon, Y., Williams, R.J.: Reflected Brownian motion in a cone with radially homogeneous reflection field. Trans. Am. Math. Soc.327, 739–780 (1991)Google Scholar
  20. 20.
    Mandelbaum, A.: The dynamic complementarity problem. Math. Oper. Res. (to appear)Google Scholar
  21. 21.
    Mandelbaum, A., Van der Heyden, L.: Complementarity and reflection (unpublished work, 1987)Google Scholar
  22. 22.
    Reiman, M.I.: Open queueing networks in heavy traffic. Math. Oper. Res.9, 441–458 (1984)Google Scholar
  23. 23.
    Reiman, M.I., Williams, R.J.: A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Relat. Fields77, 87–97 (1988);80 633 (1989)Google Scholar
  24. 24.
    Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1991Google Scholar
  25. 25.
    Sharpe, M.J.: General theory of Markov processes. Boston: Academic Press 1988Google Scholar
  26. 26.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • L. M. Taylor
    • 1
  • R. J. Williams
    • 2
  1. 1.Department of MathematicsCalifornia State UniversitySacramentoUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

Personalised recommendations