α-Wiener Bridges: Singularity of Induced Measures and Sample Path Properties

Abstract

Let us consider the process given by the SDE , t ∊ [0, T), where α ∊ ℝ, T ∊ (0, ∞), and (B t ) 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 ≠ 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 as t ↑ T.

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