Let {W(t);t ≧ 0} be
the standard Wiener process. The probabilities P[sup0≦t≦TW(t) ≧ b] and
P[sup0≦t≦TW(t) −at ≧ b] are well known. This paper gives the probabilities of the
type P[sup0≦t≦TW(t) −f(t) ≧ b] for a large class of differentiable functions f(t) by
the use of integral equation techniques.