Karhunen-Loève Expansion for the Second Order Detrended Brownian Motion

Abstract

Based on the norm in the Hilbert Space $L^2[0,1]$, the second order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the space spanned by nonlinear function subspace. Karhunen-Loève expansion for this process is obtained together with the relationship of that of a generalized Brownian bridge. As applications, Laplace transform, large deviation, and small deviation are given.