alpha-Wiener bridges: singularity of induced measures and sample path properties

Abstract

Let us consider the process $(X_t^{(\alpha)})_{t\in[0,T)}$ given by the SDE $dX_t^{(\alpha)} = -\frac{\alpha}{T-t}X_t^{(\alpha)} dt+ dB_t$, $t\in[0,T)$, where $\alpha\in R$, $T\in(0,\infty)$, and $(B_t)_{t\geq 0}$ is a standard Wiener process. In case of $\alpha>0$ the process $X^{(\alpha)}$ is known as an $\alpha$-Wiener bridge, in case of $\alpha=1$ as the usual Wiener bridge. We prove that for all $\alpha,\beta\in R$, $\alpha\ne\beta$, the probability measures induced by the processes $X^{(\alpha)}$ and $X^{(\beta)}$ are singular on C[0,T). Further, we investigate regularity properties of $X_t^{(\alpha)}$ as $t\uparrow T$.