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Levels of crossing probability for Brownian motion
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Dobromir P. Kralchev
Published/Copyright:
May 19, 2008
Abstract
Let (Bs)s≥t be Brownian motion, t be the starting moment, Bt = x, EBs = x, DBs = s – t. Let T ≥ t be a fixed time-horizon, and lower(s), upper(s) be two smooth real functions, defined for s ∈ [t; T], such that lower(s) < upper(s) for all s ∈ [t; T), lower(t) ≤ x ≤ upper (t). Finally, let τ = inf {s ∈ [t; T)| Bs = lower(s) or Bs = upper(s)}, where inf ø = T, and let 
= {Bτ = upper(τ)}, 
= {Bτ = lower(τ)}, 
. We obtain explicit formulae for Pt, x(), Pt, x (), and Pt, x (
) in two special cases: square-root boundaries with a natural horizon and arcsine boundaries with no horizon.
Key words.: Brownian motion; hitting time
Received: 2008-01-15
Published Online: 2008-05-19
Published in Print: 2008-April
© de Gruyter 2008
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