A decomposition of the Brownian path

Ioannis Karatzas and Steven E. Shreve

Statistics & Probability Letters, 1987, vol. 5, issue 2, 87-93

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.

Keywords: 60J65; 60G17; Brownian; motion; and; bridge; local; time; occupation; time; conditioning; path; decomposition (search for similar items in EconPapers)
Date: 1987
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