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A decomposition of the Brownian path

Author

Listed:
  • Karatzas, Ioannis
  • Shreve, Steven E.

Abstract

The Brownian path {[omega](s); 0 [less-than-or-equals, slant] s [less-than-or-equals, slant] t} is dissected and then reassembled in such a way that (i) the last visit [gamma]t at the origin, as well as the fragment {[omega](s); [gamma]t [less-than-or-equals, slant] s [less-than-or-equals, slant] t}, are left invariant; (ii) on [0, [gamma]t], local time becomes maximum-to-date and occupation time ofR+ becomes location of maximum; and (iii) the resulting process is again Brownian. Characterizations of conditional processes are employed to establish the result. Several consequences of the latter are discussed.

Suggested Citation

  • Karatzas, Ioannis & Shreve, Steven E., 1987. "A decomposition of the Brownian path," Statistics & Probability Letters, Elsevier, vol. 5(2), pages 87-93, March.
    • Handle: RePEc:eee:stapro:v:5:y:1987:i:2:p:87-93
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