Karhunen-Loève expansions of \(\alpha\)-Wiener bridges. (English) Zbl 1228.60047
The authors find a Karhunen-Loève expansion as well as a weighted Karhunen-Loève expansion of the process \(X^{(\alpha)}_t= \int^t_0 [(T- t)(T- s)]^\alpha dB_s\), \(t\in [0,T]\), \(\alpha> 0\), called an \(\alpha\)-Wiener bridge (or \(\alpha\)-Brownian bridge), where \(B_t\) is a standard Wiener process. This process, suitable for some econometric models, is a solution of the stochastic differential equation \(dX^{(\alpha)}_t= -{\alpha\over T-t} X^{(\alpha)}_t dt+ dB_t\) and was studied, among others, by R. Mansuy [J. Theor. Probab. 17, No. 4, 1021–1029 (2004; Zbl 1063.60049)]. Special cases \(\alpha= 1\) (usual Brownian bridge) and a \(\alpha\downarrow 0\) (standard Wiener process) are derived as corollaries. As applications, Laplace transform, series representation of tail probabilities, large and small deviations theorems for \(\int^t_0 [X^{(\alpha)}_t]^2\,dt\) are obtained.
Reviewer: Tadeusz Inglot (Wrocław)
MSC:
| 60G15 | Gaussian processes |
| 60G12 | General second-order stochastic processes |
| 60F10 | Large deviations |
| 60J65 | Brownian motion |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
\(\alpha \)-Wiener bridge; scaled Brownian bridge; Karhunen-Loève expansion; Laplace transform; large deviation; small deviationCitations:
Zbl 1063.60049References:
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