Karhunen-Loève expansion for the second order detrended Brownian motion. (English) Zbl 1469.60257

Summary: 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.


60J65 Brownian motion
60G07 General theory of stochastic processes
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