Brownian motion in a wedge with variable reflection: existence and uniqueness
R. Dante DeBlassie
Source: Ann. Probab. Volume 24, Number 1
(1996), 148-181.
Abstract
Existence and uniqueness in law of reflecting Brownian motion in a wedge is proved. The direction of reflection along the sides of the wedge varies in a reasonable fashion, except perhaps at the corner.
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1042644711
Mathematical Reviews number (MathSciNet): MR1387630
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Zentralblatt MATH identifier: 0866.60067
References
BENSOUSSAN, A. and LIONS, J. L. 1982. Controle Impulsionnel et Inequations Quasi-Variation ´ nelles. Dunod, Paris.Z.
Mathematical Reviews (MathSciNet): MR84g:49012
BURDZY, K. and MARSHALL, D. 1992. Hitting a boundary point with reflected Brownian motion. Seminaire de Probabilites XXVI. Lecture Notes in Math. 1526 81 94. Springer, New ´ ´ York. Z.
DEBLASSIE, R. D. 1990. Explicit semimartingale representation of Brownian motion in a wedge. Stochastic Processes. Appl. 34 67 97. Z.
Mathematical Reviews (MathSciNet): MR91h:60091
Zentralblatt MATH: 0694.60076
Digital Object Identifier: doi:10.1016/0304-4149(90)90057-Y
DEBLASSIE, R. D. and TOBY, E. H. 1993. Reflecting Brownian motion in a cusp. Trans. Amer. Math. Soc. 339 297 321. Z.
Zentralblatt MATH: 0791.60070
Mathematical Reviews (MathSciNet): MR1149119
Digital Object Identifier: doi:10.2307/2154220
JSTOR: links.jstor.org
DONOGHUE, W. F. 1974. Monotone Matrix Functions and Analy tic Continuation. Springer, Berlin. Z.
Mathematical Reviews (MathSciNet): MR58:6279
Zentralblatt MATH: 0278.30004
DUPUIS, P. and ISHII, H. 1991. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics Stochastics Rep. 35 31 62. Z.
Mathematical Reviews (MathSciNet): MR93e:60110
Zentralblatt MATH: 0721.60062
DUPUIS, P. and ISHII, H. 1993. SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 554 580. Z. N
Mathematical Reviews (MathSciNet): MR94c:60128
Zentralblatt MATH: 0787.60099
Digital Object Identifier: doi:10.1214/aop/1176989415
Project Euclid: euclid.aop/1176989415
EL KAROUI, N. 1975. Processus de reflexion sur. Seminaire de Probabilites IX. Lecture Notes ´ ´ ´ in Math. 465 534 554. Springer, Berlin. Z.
EL KAROUI, N. and CHALEy AT-MAUREL, M. 1978. Un probleme de reflexion au temps local et aux ´ equations differentielles stochastiques sur, cas continu, temps locaux. Asterisque ´ ´ ´ 52-53 117 144. Z.
EL KAROUI, N., CHALEy AT-MAUREL, M. and MARCHAL, B. 1980. Reflexion discontinue et sy stemes ´ stochastiques. Ann. Probab. 8 1049 1067. Z.
HALL, P. and HEy DE, C. C. 1980. Martingale Limit Theory and Its Applications. Academic Press, New York. Z.
Mathematical Reviews (MathSciNet): MR624435
Zentralblatt MATH: 0462.60045
HARRISON, J. M. and REIMAN, M. I. 1981. Reflected Brownian motion on an orthant. Ann. Probab. 9 302 308.Z.
Mathematical Reviews (MathSciNet): MR82c:60141
Zentralblatt MATH: 0462.60073
Digital Object Identifier: doi:10.1214/aop/1176994471
Project Euclid: euclid.aop/1176994471
IKEDA, N. and WATANABE, S. 1989. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. Z.
Mathematical Reviews (MathSciNet): MR1011252
KWON, Y. 1992. The submartingale problem for Brownian motion in a cone with non-constant oblique reflection. Probab. Theory Related Fields 92 351 391. Z.
Mathematical Reviews (MathSciNet): MR93g:60174
Digital Object Identifier: doi:10.1007/BF01300561
KWON, Y. and WILLIAMS, R. J. 1991. Reflected Brownian motion in a cone with radially homogeneous reflection field. Trans. Amer. Math. Soc. 327 739 780. Z.
Mathematical Reviews (MathSciNet): MR92a:60174
Zentralblatt MATH: 0742.60075
Digital Object Identifier: doi:10.2307/2001821
JSTOR: links.jstor.org
LIONS, P. L. and SZNITMAN, A. S. 1984. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511 537. Z.
Zentralblatt MATH: 0598.60060
Mathematical Reviews (MathSciNet): MR745330
Digital Object Identifier: doi:10.1002/cpa.3160370408
MCKEAN, H. P. 1963. A Skorokhod's integral equation for a reflecting barrier diffusion. J. Math. Ky oto Univ. 3 86 88. Z.
Mathematical Reviews (MathSciNet): MR157406
Zentralblatt MATH: 0202.46601
Project Euclid: euclid.kjm/1250524858
MCKEAN, H. P. 1969. Stochastic Integrals. Academic Press, New York. Z.
Mathematical Reviews (MathSciNet): MR40:947
REVUZ, D. and YOR, M. 1991. Continuous Martingales and Brownian Motion. Springer, Berlin. Z.
Mathematical Reviews (MathSciNet): MR92d:60053
ROGERS, L. C. G. 1990. Brownian motion in a wedge with variable skew reflection: II. In Z. Diffusion Processes and Related Problems in Analy sis M. A. Pinsky, ed. 1 95 115. Birkhauser, Boston. ¨ Z.
ROGERS, L. C. G. 1991. Brownian motion in a wedge with variable skew reflection. Trans. Amer. Math. Soc. 326 227 236.
Zentralblatt MATH: 0748.60074
Mathematical Reviews (MathSciNet): MR1008701
Digital Object Identifier: doi:10.2307/2001862
JSTOR: links.jstor.org
ROGERS, L. C. G. and WILLIAMS, D. 1987. Diffusions, Markov Processes and Martingales 2. Wiley, New York. Z.
Mathematical Reviews (MathSciNet): MR921238
Zentralblatt MATH: 0977.60005
SAISHO, Y. 1987. Stochastic differential equations for multi-dimensional domains with reflecting boundary. Probab. Theory Related Fields 74 455 477. Z.
Mathematical Reviews (MathSciNet): MR88b:60139
Digital Object Identifier: doi:10.1007/BF00699100
SKOROKHOD, A. V. 1961. Stochastic equations for diffusion processes in a bounded region 1. Teor. Veroy atnost. i Primenen. 6 264 274. Z.
SKOROKHOD, A. V. 1962. Stochastic equations for diffusion processes in a bounded region 2. Teor. Veroy atnost. i Primenen. 7 3 23. Z.
STROOCK, D. W. and VARADHAN, S. R. S. 1971. Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 147 225. Z.
Mathematical Reviews (MathSciNet): MR43:2774
Zentralblatt MATH: 0227.76131
Digital Object Identifier: doi:10.1002/cpa.3160240206
TANAKA, H. 1979. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 163 177. Z.
Zentralblatt MATH: 0423.60055
Mathematical Reviews (MathSciNet): MR529332
Project Euclid: euclid.hmj/1206135203
TSUCHIy A, M. 1976. On the SDE for a Brownian motion with oblique reflection on the half plane. In Proceedings of the International Sy mposium on Stochastic Differential Z. Equations K. Ito, ed.. Wiley, New York. Z.
TSUCHIy A, M. 1980. On the SDE for a two-dimensional Brownian motion with boundary condition. J. Math. Soc. Japan 32 233 249. Z.
Mathematical Reviews (MathSciNet): MR567417
Zentralblatt MATH: 0444.60066
Digital Object Identifier: doi:10.2969/jmsj/03220233
Project Euclid: euclid.jmsj/1240234796
VARADHAN, S. R. S. and WILLIAMS, R. J. 1985. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 405 443. Z.
Mathematical Reviews (MathSciNet): MR87c:60066
Zentralblatt MATH: 0579.60082
Digital Object Identifier: doi:10.1002/cpa.3160380405
WATANABE, S. 1971. On stochastic differential equations for multidimensional diffusion processes with boundary conditions. J. Math. Ky oto Univ. 11 169 180. Z.
Mathematical Reviews (MathSciNet): MR275537
Project Euclid: euclid.kjm/1250523692
WILLIAMS, R. J. 1985a. Reflected Brownian motion wedge: semimartingale property. Z. Wahrsch. Verw. Gebiete 69 161 176. Z.
Mathematical Reviews (MathSciNet): MR779455
WILLIAMS, R. J. 1985b. Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Probab. 13 758 778. Z.
Mathematical Reviews (MathSciNet): MR799421
Zentralblatt MATH: 0596.60078
Digital Object Identifier: doi:10.1214/aop/1176992907
Project Euclid: euclid.aop/1176992907
Zy GMUND, A. 1979. Trigonometric Series 1 and 2. combined, 2nd ed. Cambridge Univ. Press.
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