We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function of the L 2[0, T]-norm square of X (α) studying also its asymptotic behavior (large and small deviation).
@article{bwmeta1.element.doi-10_2478_s11533-010-0090-8,
author = {M\'aty\'as Barczy and Endre Igl\'oi},
title = {Karhunen-Lo\`eve expansions of $\alpha$-Wiener bridges},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {65-84},
zbl = {1228.60047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0090-8}
}
Mátyás Barczy; Endre Iglói. Karhunen-Loève expansions of α-Wiener bridges. Open Mathematics, Tome 9 (2011) pp. 65-84. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0090-8/
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