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

Let us consider the process (X (α) t )t∈[0,T ) given by the SDE dX (α) t = − α T−t X (α) t dt+ dBt, t ∈ [0, T ), where α ∈ R, T ∈ (0,∞), and (Bt)t>0 is a standard Wiener process. In case of α > 0 the process X(α) is known as an α-Wiener bridge, in case of α = 1 as the usual Wiener bridge. We prove that for all α, β ∈ R, α 6= β, the probability measures induced by the processes X(α) and X(β) are singular on ( C[0, T ),B(C[0, T )) ) . Further, we investigate regularity properties of X (α) t as t ↑ T .