Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Embedding processes in Brownian motion in $ {\bf R}\sp{n}$


Author: Neil Falkner
Journal: Trans. Amer. Math. Soc. 267 (1981), 335-363
MSC: Primary 60G40; Secondary 60G17, 60J65
DOI: https://doi.org/10.1090/S0002-9947-1981-0626478-9
Erratum: Trans. Amer. Math. Soc. 272 (1982), 811.
MathSciNet review: 626478
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a potential-theoretic characterization of the right-continuous processes which can be embedded in Brownian motion in $ {{\mathbf{R}}^n}$ by means of an increasing family of standard stopping times. In general it is necessary to use a Brownian motion process whose filtration is richer than the natural one.


References [Enhancements On Off] (What's this?)

  • 1. Azéma and M. Yor, 1. Une solution simple au problème de Skorokhod, Strasbourg Séminaire de Probabilités XIII, Lecture Notes in Math., vol. 721, Springer-Verlag, Berlin, 1979. MR 544811 (82e:60091)
  • 2. R. Baxter and R. V. Chacon, 1. Potentials of stopped distributions, Illinois J. Math. 18 (1974), 649-656. MR 0358960 (50:11417)
  • 3. -, 2. Enlargement of $ \sigma $-algebras and compactness of time changes, Canad. J. Math. 29 (1977), 1055-1065. MR 0517870 (58:24524)
  • 4. M. Blumenthal and R. K. Getoor, 1. Markov processes and potential theory, Academic Press, New York, 1968. MR 0264757 (41:9348)
  • 5. V. Chacon, 1. Potential processes, Trans. Amer. Math. Soc. 226 (1977), 39-58. MR 0501374 (58:18746)
  • 6. V. Chacon and B. Jamison, 1. On a fundamental property of Markov processes with an application to equivalence under time changes, Israel J. Math. (to appear). MR 571533 (81i:60065)
  • 7. Dellacherie and P. A. Meyer, 1. Probabilités et potentiel, Edition Entièrement Refondue, Hermann, Paris, 1975. MR 0488194 (58:7757)
  • 8. E. Dubins, 1. On a theorem of Skorokhod, Ann. Math. Statist. 39 (1968), 2094-2097. MR 0234520 (38:2837)
  • 9. E. Dubins and G. Schwarz, On continuous martingales, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 913-916. MR 0178499 (31:2756)
  • 10. Falkner, 1. On Skorohod embedding in $ n$-dimensional Brownian motion by means of natural stopping times, Strasbourg Séminaire de Probabilités XIV, Lecture Notes in Math., vol. 784, Springer-Verlag, Berlin, 1980. MR 580142 (83f:60106)
  • 11. -, 2. The distribution of Brownian motion in $ {{\mathbf{R}}^n}$ at a natural stopping time, Adv. in Math. (to appear). MR 619096 (83d:60092)
  • 12. Lévy, 1. Processus stochastiques et mouvement Brownien, Gauthier-Villars, Paris, 1965. MR 0190953 (32:8363)
  • 13. A. Meyer, 1. Sur un article de Dubins, Strasbourg Séminaire de Probabilitiés V, Lecture Notes in Math., vol. 191, Springer-Verlag, Berlin, 1971. MR 0375456 (51:11649)
  • 14. -, 2. Processus de Markov, Lecture Notes in Math., vol. 26, Springer-Verlag, Berlin, 1967. MR 0219136 (36:2219)
  • 15. -, 3. Probabilités et potentiel, Hermann, Paris, 1966. MR 0205287 (34:5118)
  • 16. Monroe, 1. On embedding right continuous martingales in Brownian motion, Ann. Math. Statist. 43 (1972), 1293-1311. MR 0343354 (49:8096)
  • 17. -, 2. Processes that can be embedded in Brownian motion, Ann. Probab. 6 (1978), 42-56. MR 0455113 (56:13353)
  • 18. K. Pachl, 1. Disintegration and compact measures, Math. Scand. 43 (1978), 157-168. MR 523833 (80d:28020)
  • 19. H. Root, 1. The existence of certain stopping times of Brownian motion, Ann. Math. Statist. 40 (1969), 715-718. MR 0238394 (38:6670)
  • 20. Rost, 1. Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichem Potential, Manuscripta Math. 3 (1970), 321-330. MR 0273691 (42:8568)
  • 21. V. Sazonov, 1. On perfect measures, Amer. Math. Soc. Transl. (2) 48 (1965), 229-254.
  • 22. V. Skorokhod, 1. Studies in the theory of random processes, Addison-Wesley, Reading, Mass., 1965. MR 0185620 (32:3082b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60G40, 60G17, 60J65

Retrieve articles in all journals with MSC: 60G40, 60G17, 60J65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0626478-9
Keywords: Potential process, time change
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society