A Karhunen-Loève expansion for a mean-centered Brownian bridge

Abstract

The processes of the form YK (t) = B (t) - 6 Kt (1 - t) ∫01 B (u) d u, where K is a constant, and B (·) a Brownian bridge, are investigated. We show that Y0 (·) and Y2 (·) are both Brownian bridges, and establish the independence of Y1 (·) and ∫01 B (u) d u, this implying that the law of Y1 (·) coincides with the conditional law of B, given that ∫01 B (u) d u = 0. We provide the Karhunen-Loève expansion on [0, 1] of Y1 (·), making use of the Bessel functions J1 / 2 and J3 / 2. Applications and variants of these results are discussed. In particular, we establish a comparison theorem concerning the supremum distributions of YK′ (·) and YK″ (·) on [0, 1]. © 2007 Elsevier B.V. All rights reserved.