Permutations, signs and the Brownian bridge

Let B(t), 0 less than or equal to t less than or equal to 1 be a Brownian Bridge, and let f : [0, 1] --> {+1, -1} be a non-random, measurable function. Then for every t greater than or equal to 0 the following holds: P (max(0 less than or equal to s less than or equal to 1) \B(s)\ greater than or equal to t) less than or equal to P (max(0 less than or equal to s less than or equal to 1) \integral(0)(s) f(u) dB(u)\ greater than or equal to t) less than or equal to P (max(0 less than or equal to s less than or equal to 1) \B(s)\ greater than or equal to t/2). The result follows from a discrete-time maximal inequality for signs via weak convergence. We will present applications of this result in the area of mathematical finance. (C) 2000 Elsevier Science B.V. All rights reserved.