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Karhunen-Loève expansions of α-Wiener bridges

Mátyás BarczyEndre Iglói — 2011

Open Mathematics

We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation d X t ( α ) = - α T - t X t ( α ) d t + d B t , t [ 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...

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