Karhunen-Loève expansions of α-Wiener bridges

Open Mathematics (2011)

• Volume: 9, Issue: 1, page 65-84
• ISSN: 2391-5455

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Abstract

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We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation $d{X}_{t}^{\left(\alpha \right)}=-\frac{\alpha }{T-t}{X}_{t}^{\left(\alpha \right)}dt+d{B}_{t},t\in \left[0,T\right)$ , 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).

How to cite

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Mátyás Barczy, and Endre Iglói. "Karhunen-Loève expansions of α-Wiener bridges." Open Mathematics 9.1 (2011): 65-84. <http://eudml.org/doc/269585>.

@article{MátyásBarczy2011,
abstract = {We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation $dX\_t^\{(\alpha )\} = - \frac\{\alpha \}\{\{T - t\}\}X\_t^\{(\alpha )\} dt + dB\_t ,t \in [0,T)$ , 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).},
author = {Mátyás Barczy, Endre Iglói},
journal = {Open Mathematics},
keywords = {α-Wiener bridge; Scaled Brownian bridge; Karhunen-Loève expansion; Laplace transform; Large deviation; Small deviation; -Wiener bridge; scaled Brownian bridge; large deviation; small deviation},
language = {eng},
number = {1},
pages = {65-84},
title = {Karhunen-Loève expansions of α-Wiener bridges},
url = {http://eudml.org/doc/269585},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Mátyás Barczy
AU - Endre Iglói
TI - Karhunen-Loève expansions of α-Wiener bridges
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 65
EP - 84
AB - We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation $dX_t^{(\alpha )} = - \frac{\alpha }{{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T)$ , 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).
LA - eng
KW - α-Wiener bridge; Scaled Brownian bridge; Karhunen-Loève expansion; Laplace transform; Large deviation; Small deviation; -Wiener bridge; scaled Brownian bridge; large deviation; small deviation
UR - http://eudml.org/doc/269585
ER -

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