Advertisement

Set-indexed Brownian Motion on Increasing Paths

  • Ely Merzbach
  • Arthur Yosef
Article
  • 45 Downloads

Abstract

We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projections on all the strict increasing continuous paths are one-parameter time-change Brownian motions. We present some applications.

Keywords

Set-indexed process Brownian motion Increasing path 

Mathematics Subject Classification (2000)

60G15 60G48 60G60 

References

  1. 1.
    Aletti, G., Saada, D.: Set-indexed Ito calculus along paths. Stoch. Anal. Appl. 22(4), 1027–1066 (2004) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Math. 134, 111–183 (1975) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dalang, R.C.: Level sets and excursions of Brownian sheet. In: Capasso, V., Ivanoff, B.G., Dalang, R.C., Merzbach, E., Dozzi, M., Mountford, T.S. (eds.) Topics in Spatial Stochastic Processes. Lecture Notes in Mathematics, vol. 1802, pp. 167–208. Springer, Berlin (2001) Google Scholar
  4. 4.
    Durrett, R.: Brownian Motion and Martingales in Analysis. The Wadsworth Mathematics Series, Belmont, 1–43, 1971 Google Scholar
  5. 5.
    Herbin, E., Merzbach, E.: A characterization of the set-indexed Brownian motion by increasing paths. C. R. Acad. Sci. Paris, Sec. 1 343, 767–772 (2006) MATHMathSciNetGoogle Scholar
  6. 6.
    Ivanoff, G., Merzbach, E.: Set-Indexed Martingales, Monographs on Statistics and Applied Probability. Chapman and Hall/CRC, London (1999) Google Scholar
  7. 7.
    Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields. Springer, London (2002) MATHGoogle Scholar
  8. 8.
    Merzbach, E., Nualart, D.: Different kinds of two parameter martingales. Isr. J. Math. 52(3), 193–207 (1985) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, New York (1991) MATHGoogle Scholar
  10. 10.
    Walsh, J.B.: Optional Increasing Paths. Lecture Notes in Mathematics, vol. 863, pp. 172–201. Springer, Berlin (1980) Google Scholar
  11. 11.
    Zakai, M.: Some classes of two-parameter martingales. Ann. Probab. 9, 255–265 (1981) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

Personalised recommendations