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 Electronic Journal of Probability > Vol. 11 (2006) > Paper 36 open journal systems 


Reflected diffusions defined via the extended Skorokhod map

Kavita Ramanan, Carnegie Mellon University


Abstract
This work introduces the extended Skorokhod problem (ESP) and associated extended Skorokhod map (ESM) that enable a pathwise construction of reflected diffusions that are not necessarily semimartingales. Roughly speaking, given the closure G of an open connected set in R J, a non-empty convex cone d(x) in R J, specified at each point x on the boundary of G, and a cadlag trajectory ψ taking values in R J, the ESM defines a constrained version φ of ψ that takes values in G and is such that the increments of φ - ψ on any interval [s,t] lie in the closed convex hull of the directions d(φ(u)), u ∈ (s,t]. General deterministic properties of the ESP are first established under the only assumption that the graph of d(.) is closed. Next, for a class of multi-dimensional ESPs on polyhedral domains, pathwise uniqueness and existence of strong solutions to the associated stochastic differential equations is established. In addition, it is also proved that these reflected diffusions are semimartingales on [0,τ0], where τ0 is the time to hit the set of points x on the boundary for which d(x) contains a line. One motivation for the study of this class of reflected diffusions is that they arise as approximations of queueing networks in heavy traffic that use the so-called generalised processor sharing discipline.


Full text: PDF

Pages: 934-992

Published on: October 18, 2006
Bibliography
  1. Anderson, Robert F.; Orey, Steven. Small random perturbation of dynamical systems with reflecting Nagoya Math. J. 60 (1976), 189--216. MR0397893 (53 #1749)
  2. Aubin, Jean-Pierre; Frankowska, Hélène. Set-valued analysis. Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. xx+461 pp. ISBN: 0-8176-3478-9 MR1048347 (91d:49001)
  3. Bichteler, Klaus. Stochastic integration and $Lsp{p}$-theory of semimartingales. Ann. Probab. 9 (1981), no. 1, 49--89. MR0606798 (82g:60071)
  4. Bernard, Alain; el Kharroubi, Ahmed. Régulations déterministes et stochastiques dans le premier Rsp n$] Stochastics Stochastics Rep. 34 (1991), no. 3-4, 149--167. MR1124833 (92k:60178)
  5. Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396 (38 #1718)
  6. Borodin, Andrei N.; Salminen, Paavo. Handbook of Brownian motion---facts and formulae. Probability and its Applications. Birkhäuser Verlag, Basel, 1996. xiv+462 pp. ISBN: 3-7643-5463-1 MR1477407 (98i:60077)
  7. V. Burenkov. Sobolev Spaces on Domains. Teubner-Texte zur Mathematik, Leipzig, 1998.
  8. Burdzy, Krzysztof; Toby, Ellen. A Skorohod-type lemma and a decomposition of reflected Brownian Ann. Probab. 23 (1995), no. 2, 586--604. MR1334162 (96i:60045)
  9. Chen, Zhen Qing. On reflected Dirichlet spaces. Probab. Theory Related Fields 94 (1992), no. 2, 135--162. MR1191106 (93m:31014)
  10. Chen, Zhen Qing. On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Related Fields 94 (1993), no. 3, 281--315. MR1198650 (94b:60090)
  11. Chen, Hong; Mandelbaum, Avi. Discrete flow networks: bottleneck analysis and fluid Math. Oper. Res. 16 (1991), no. 2, 408--446. MR1106809 (92b:60090)
  12. Costantini, C.. The Skorohod oblique reflection problem in domains with corners and Probab. Theory Related Fields 91 (1992), no. 1, 43--70. MR1142761 (93e:60109)
  13. J. Dai and R. Williams. Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons. Theor. Probab. Appl., 50:3-53, 1995.
  14. DeBlassie, R. Dante. Explicit semimartingale representation of Brownian motion in a Stochastic Process. Appl. 34 (1990), no. 1, 67--97. MR1039563 (91h:60091)
  15. DeBlassie, R. Dante; Toby, Ellen H. Reflecting Brownian motion in a cusp. Trans. Amer. Math. Soc. 339 (1993), no. 1, 297--321. MR1149119 (93k:60199)
  16. DeBlassie, R. Dante; Toby, Ellen H. On the semimartingale representation of reflecting Brownian motion in a Probab. Theory Related Fields 94 (1993), no. 4, 505--524. MR1201557 (94a:60118)
  17. Doob, J. L. Measure theory. Graduate Texts in Mathematics, 143. Springer-Verlag, New York, 1994. xii+210 pp. ISBN: 0-387-94055-3 MR1253752 (95c:28001)
  18. Dupuis, Paul; Ishii, Hitoshi. On Lipschitz continuity of the solution mapping to the Skorokhod Stochastics Stochastics Rep. 35 (1991), no. 1, 31--62. MR1110990 (93e:60110)
  19. Dupuis, Paul; Ishii, Hitoshi. On oblique derivative problems for fully nonlinear second-order Hokkaido Math. J. 20 (1991), no. 1, 135--164. MR1096165 (92b:35060)
  20. Kushner, Harold J.; Dupuis, Paul G. Numerical methods for stochastic control problems in continuous Applications of Mathematics (New York), 24. Springer-Verlag, New York, 1992. x+439 pp. ISBN: 0-387-97834-8 MR1217486 (94e:93005)
  21. Dupuis, Paul; Ramanan, Kavita. A Skorokhod problem formulation and large deviation analysis of a Queueing Systems Theory Appl. 28 (1998), no. 1-3, 109--124. MR1628485 (99c:60201)
  22. Dupuis, Paul; Ramanan, Kavita. Convex duality and the Skorokhod problem. I, II. Probab. Theory Related Fields 115 (1999), no. 2, 153--195, 197--236. MR1720348 (2001f:49041)
  23. Dupuis, Paul; Ramanan, Kavita. Convex duality and the Skorokhod problem. I, II. Probab. Theory Related Fields 115 (1999), no. 2, 153--195, 197--236. MR1720348 (2001f:49041)
  24. Dupuis, Paul; Ramanan, Kavita. A multiclass feedback queueing network with a regular Skorokhod Queueing Syst. 36 (2000), no. 4, 327--349. MR1823974 (2002g:60143)
  25. Freidlin, Mark. Functional integration and partial differential equations. Annals of Mathematics Studies, 109. Princeton University Press, Princeton, NJ, 1985. x+545 pp. ISBN: 0-691-08354-1; 0-691-08362-2 MR0833742 (87g:60066)
  26. Fukushima, Masatoshi; =Oshima, Y=oichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-X MR1303354 (96f:60126)
  27. Harrison, J. Michael; Reiman, Martin I. Reflected Brownian motion on an orthant. Ann. Probab. 9 (1981), no. 2, 302--308. MR0606992 (82c:60141)
  28. Holmes, Richard B. Smoothness of certain metric projections on Hilbert space. Trans. Amer. Math. Soc. 184 (1973), 87--100. MR0326252 (48 #4596)
  29. Kallenberg, Olav. Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, 1997. xii+523 pp. ISBN: 0-387-94957-7 MR1464694 (99e:60001)
  30. Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1988. xxiv+470 pp. ISBN: 0-387-96535-1 MR0917065 (89c:60096)
  31. Kwon, Y.; Williams, R. J.. Reflected Brownian motion in a cone with radially homogeneous Trans. Amer. Math. Soc. 327 (1991), no. 2, 739--780. MR1028760 (92a:60174)
  32. Lions, P.-L.; Sznitman, A.-S. Stochastic differential equations with reflecting boundary Comm. Pure Appl. Math. 37 (1984), no. 4, 511--537. MR0745330 (85m:60105)
  33. A. Mandelbaum and A. Van der Heyden. Complementarity and reflection. Unpublished work, 1987.
  34. Moreau, Jean-Jacques. Proximitéet dualité dans un espace hilbertien. (French) Bull. Soc. Math. France 93 1965 273--299. MR0201952 (34 #1829)
  35. Parthasarathy, K. R. Probability measures on metric spaces. Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London 1967 xi+276 pp. MR0226684 (37 #2271)
  36. Ramanan, Kavita; Reiman, Martin I. Fluid and heavy traffic diffusion limits for a generalized processor Ann. Appl. Probab. 13 (2003), no. 1, 100--139. MR1951995 (2004b:60220)
    37. K. Ramanan and M.I. Reiman. The heavy traffic limit of an unbalanced generalized processor sharing model. Preprint, 2005.
  37. Reiman, M. I.; Williams, R. J.. A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 (1988), no. 1, 87--97. MR0921820 (89a:60191)
  38. Rockafellar, R. Tyrrell. Convex analysis. Princeton Mathematical Series, No. 28 Princeton University Press, Princeton, N.J. 1970 xviii+451 pp. MR0274683 (43 #445)
  39. Royden, H. L. Real analysis. The Macmillan Co., New York; Collier-Macmillan Ltd., London 1963 xvi+284 pp. MR0151555 (27 #1540)
  40. Saisho, Yasumasa. Stochastic differential equations for multidimensional domain with Probab. Theory Related Fields 74 (1987), no. 3, 455--477. MR0873889 (88b:60139)
  41. A.V. Skorokhod. Stochastic equations for diffusions in a bounded region. Theor. of Probab. and its Appl., 6:264--274, 1961.
  42. Stroock, Daniel W.; Varadhan, S. R. S. Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 1971 147--225. MR0277037 (43 #2774)
  43. Tanaka, Hiroshi. Stochastic differential equations with reflecting boundary condition in Hiroshima Math. J. 9 (1979), no. 1, 163--177. MR0529332 (80k:60075)
  44. Taylor, L. M.; Williams, R. J.. Existence and uniqueness of semimartingale reflecting Brownian motions Probab. Theory Related Fields 96 (1993), no. 3, 283--317. MR1231926 (94m:60161)
  45. Varadhan, S. R. S.; Williams, R. J.. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1985), no. 4, 405--443. MR0792398 (87c:60066)
  46. Whitt, Ward. Stochastic-process limits. Springer Series in Operations Research. Springer-Verlag, New York, 2002. xxiv+602 pp. ISBN: 0-387-95358-2 MR1876437 (2003f:60005)
  47. Williams, R. J.. Reflected Brownian motion in a wedge: semimartingale property. Z. Wahrsch. Verw. Gebiete 69 (1985), no. 2, 161--176. MR0779455 (86h:60164)
  48. Williams, R. J.. Semimartingale reflecting Brownian motions in the orthant. 125--137, IMA Vol. Math. Appl., 71, Springer, New York, 1995. MR1381009 (96k:60213)
















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Electronic Journal of Probability. ISSN: 1083-6489