Karhunen-Loève expansions of α-Wiener bridges

We study Karhunen-Loeve 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-Loeve 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).