Karhunen-Loeve expansions of alpha-Wiener bridges

Abstract

We study Karhunen-Loeve expansions of the process \((X_t^{(\alpha)})_{t\in[0,T)}\) given by the stochastic differential equation \(dX_t^{(\alpha)} = -\frac\alpha{T-t} X_t^{(\alpha)} dt+ dB_t,\) \(t\in[0,T),\) with an initial condition \(X_0^{(\alpha)}=0,\) where \(\alpha>0,\) \(T\in(0,\infty)\) and \((B_t)_{t\geq 0}\) is a standard Wiener process. This process is called an \(\alpha\)-Wiener bridge or a scaled Brownian bridge, and in the special case of \(\alpha=1\) the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of \(X^{(\alpha)}\). As applications, we calculate the Laplace transform and the distribution function of the \(L^2[0,T]\)-norm square of \(X^{(\alpha)}\) studying also its asymptotic behavior (large and small deviation).